Optics & Laser Technology 157 (2023) 108673
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Optics and Laser Technology
journal homepage: www.elsevier.com/locate/optlastec
Full length article
Array of photonic hooks generated by multi-dielectric structure
Yu-Jing Yang, De-Long Zhang *, Ping-Rang Hua *
Department of Opto-electronics and Information Engineering, School of Precision Instruments and Opto-electronics Engineering, and Key Laboratory of Optoelectronic
Information Science & Technology (Ministry of Education), Tianjin University, Tianjin 300072, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
Array of photonic hooks
Multi–dielectric structure
Optical inhomogeneity
Generation of array of photonic hooks (PHs) by a multi–dielectric structure that consists of periodically arrayed
scatterer units has been systematically studied. The study focuses on the effects of geometric and optical pa­
rameters of the multi–dielectric structure on the characteristics of generated PHs, especially their bending angles.
These geometric and optical parameters include size, shape, refractive index and number of scatterer units, and
their adjacent spacing. The results show that the spacing, size and refractive index play a predominant role in
tailoring the performance of the PHs array, and the spacing affects mainly the characteristics of a pair of sym­
metrical PHs while the size and refractive index differences between scatterer units affect mainly those of
asymmetrical PHs as a result of the introduction of optical inhomogeneity. In the aspect of scatterer shape, for a
multi–dielectric structure that consists of some cylinders, as the light wave is incident parallel (perpendicular) to
the cylinder axis, its scattering characteristics are similar to those of the multi–dielectric structure that consists of
some cuboids (spheres). In particular, for a structure consisted of cylinders or cuboids with long length in the z
direction, an intensive photonic nanojet is additionally generated and located between the paired PHs provided
that the wave is incident perpendicular to the cylinder axis or the longest side of cuboid. In the aspect of scatterer
unit number, the multi–dielectric structures of odd and even scatterer units generate PHs with different char­
acteristics. The paired PHs generated by multi–dielectric structure with equal spacings always bend outward.
However, the paired PHs may bend inward provided that the multi–dielectric structure is designed properly. We
have also studied the features of 2D PHs array. The results show that the 2D PHs array displays similar features to
the 1D case. The application potential of both 1D and 2D PHs arrays is evaluated.
1. Introduction
Photonic nanojet (PNJ) refers to a light spot produced by focusing of
wave scattered by a particle with a wavelength scale and located behind
the scatterer. It is featured by high light intensity, sub-wavelength waist
and wavelength-scale propagation distance experienced a weak
diffraction [1–3]. Due to these excellent performance parameters, the
PNJ exhibits great application potential in fields of high-resolution im­
aging, nano-particle detection, and the enhancement of backscattering
and Raman scattering [4–8]. To expand the application of PNJ in par­
ticle trapping and manipulation, Minin brothers proposed firstly the
conception of photonic hook (PH) based on the formation physical
principle of the PNJ in 2015 [9–11]. The PH refers to a special PNJ
featured by a curved light spot and the curved spot is formed by phase
modulation of scattered wave, which leads to the deformation of
wavefront and hence the curvature of light beam. With a hook shape and
the above-mentioned excellent properties of PNJ, PH may find its
applications in the fields of optical-mechanical manipulation, nano­
particles trapping, optical switch, and image contrast enhancement
[12–19].
The PH is usually produced by a dielectric particle with broken
symmetry and integrity. In specific, the PH is obtained by breaking the
optical homogeneity of particle and rendering the isophase plane of
scattered wave curved in different extents. A particle structure with
broken symmetry is mainly obtained by following two ways. One way is
to design a structure with a fixed refractive index distribution while an
asymmetrical geometry, for example, appending a triangular prism to
the front of a cuboid [20–22], coating a metal film onto part of a
microcylinder [23]. The other way is to design a structure with a sym­
metrical geometry while an inhomogeneous refractive index distribu­
tion, such as a Janus particle [24–27], a cuboid cladded with a cylinder
or a hexagonal prism [28–29]. In addition, a particle with a symmetrical
structure has also been demonstrated to produce a PH by using a
structured light beam irradiation [30,3]. The essence of this method lies
* Corresponding authors.
E-mail addresses: dlzhang@tju.edu.cn (D.-L. Zhang), prhua@tju.edu.cn (P.-R. Hua).
https://doi.org/10.1016/j.optlastec.2022.108673
Received 24 July 2022; Received in revised form 5 September 2022; Accepted 6 September 2022
Available online 14 September 2022
0030-3992/© 2022 Elsevier Ltd. All rights reserved.
Y.-J. Yang et al.
Optics and Laser Technology 157 (2023) 108673
in the replacement of the optical inhomogeneity of structure with the
asymmetry of incident beam. Minin brothers firstly proposed and
experimentally verified the idea on the basis of a symmetrical scatterer
partially irradiated by a plane wave [30]. Then, Gu et al. performed a
further theoretical study and demonstrated that the characteristics of pH
can be tailored by changing irradiation condition [3].
It is worthwhile to point out that the above-reviewed studies
concentrate on generation of single PH. However, an array of multiple
PHs is required for some special application scenarios, such as batch
manipulation of multiple nanoparticles and biological cells, simulta­
neous sensing and tracking of multiple particles, photolithography and
preparation of special array patterns, control of multi-node optical
switches, and on-chip connection of multiple paired electrodes. To date,
only one relevant paper could be found in the previous literature as far
as we know. In 2020, Shen et al. reported twin-PHs generated by a twinellipse microcylinder [31]. However, the twin-ellipse microcylinder
scatterer adopted has a complicated structure, increasing the difficulty
of its implementation. In this paper, we firstly carried out a systematic
investigation on generation of an array of PHs by a multi–dielectric
structure that consists of periodically or aperiodically arranged scatterer
units. The simulation results show that arrays of PHs bent outward or
inward in different extents can be generated by specifically designed
multi–dielectric structures. The arrays of PHs may find applications in
above-mentioned special scenarios. And the multi–dielectric structure
scatterer possesses a number of merits, including structural simplicity
and compactness, greater scalability, and ease to fabrication, larger
flexibility in structural adjustment, and dynamic control in real time.
It is worthy of mentioning that researchers have studied scattering
characteristics of a dual-dielectric structure [32,33], which is a one kind
of multi-dielectric structure considered in present study. However,
either a usual PNJ or a single PH is generated by the studied dualdielectric structure.
PNJ formed by the interferences of incident and scattered lights is
generated behind the scatterer. The PNJ is characterized by the
following three parameters [36]: (1) focal length (FL), defined as the
distance along y-axis from the position of maximum intensity to that of
the right end face of scatterer; (2) FWHMx(z), defined as full width at half
maximum intensity along the transversal x(z)-axis at focusing point as
indicated in Fig. 1(a); (3) light intensity enhancement factor I/I0,
defined as the ratio of light intensity I at the focus point in the case of
scatterer presence to that in the case of scatterer absence I0. In Fig. 1(b),
the dual-dielectric structure is composed of two PDMS cylinders shown
in Fig. 1(a). The left and right cylinder also have geometrical parameters
of radius Rleft, Rright and length Hleft, Hright. The spacing between two
adjacent cylinders along x-axis is defined as D. Besides three parameters
to characterize PNJ, the PH is also depicted by the fourth parameter: (4)
Bending angle (θ), defined as the angle between two intersecting lines,
named line 1 and line 2 as indicated in Fig. 1(b). Line 1 is defined by the
starting point and the inflection point, and the line 2 is defined by the
inflection point and the end point [3]. The starting point is defined as the
position on the right end surface of the scatterer where the light intensity
maximizes, the inflection point is just the focusing point, and the end
point, which is indicated in Fig. 1(b), is defined as the position where the
intensity is 1∕e2 of the maximum intensity Imax and the y coordinate has
the largest value. As indicated in Fig. 1(b), this dual-dielectric structure
produces a pair of PHs with left and right bending angles of θleft and
θright.
A three-dimensional simulation is performed with the aid of a com­
mercial software CST Microwave studio. A transient solver in timedomain was adopted, and a hexahedral mesh with an element size of
one-fifteenth wavelength (λ/15) was used for the whole computational
domain [in the special case that the dual-dielectric structure has a
smaller spacing D = 0.004λ, a smaller mesh size of one-fiftieth wave­
length (λ/50) was adopted]. The open boundary conditions, usually
used in the calculation of scattered fields, were set in three directions. It
means that the surrounding medium is an infinite air space without
reflection. A continuous plane wave with the wavelength of λ = 500 nm
was selected as light source, and the adopted PDMS has a refractive
index of 1.4.
2. Structural model, principle and numerical method
The structure of single cylinder scatterer is firstly exhibited in Fig. 1.
It has an equal length of radius R, length H, and incident wavelength λ
(R = H = λ). A cartesian coordinate system is fixed on the right end face
of cylinder. With the axis of cylinder parallel to the direction of incident
light along y-axis, this cylinder is also called unflipped cylinder here­
after; while the cylinder with its axis perpendicular to incidence direc­
tion is called flipped cylinder. The cylinder adopts the polymeric
material polydimethylsiloxane (PDMS) as it has a weak absorption in
visible region and can be easily fabricated into different shaped struc­
tures with high fidelity using the modern micro/nano fabrication tech­
nology, such as replica molding process [30,34,35]. The scatterer
structure is immersed in air medium, and the ratio of refractive indices
of scatterer material and surrounding medium is defined as the refrac­
tive index ratio n. Under the illumination of a z-polarized plane wave, a
3. Numerical results and discussion
According to the number of scatterer units in multi-dielectric struc­
ture, the studied cases can be divided into two types: (1) the structure of
even scatterer units; (2) the structure of odd scatterer units. In the first
case, the dual-dielectric structure is studied as an example; and in the
second case, the tri-dielectric structure is selected as a sample.
3.1. Effect of spacing D
The attention is firstly paid to the dual-dielectric structure consisted
Fig. 1. Structural model of light scattering of (a) a single cylinder, and (b) a multi-dielectric structure that consists of two cylinders.
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Optics and Laser Technology 157 (2023) 108673
of two same microcylinders. Fig. 2(a) gives the contours of light in­
tensity of paired PHs generated by this structure in the case that spacing
varies in the range of 0.004λ-5λ. It can be seen that distinct from the case
of single scatterer, the dual-dielectric structure generates a pair of
symmetrical PHs bent outward.
One may query whether the curvature of light beam is caused by the
effect of boundary conditions on the side border of scatterer or not. To
clarify this point, we have done simulation in the case that the spacing is
fixed at 0.004λ while the distance between the side of scatterer and the
border of simulation window is increased. The result of the contour of
light intensity of PHs is shown in Fig. 2(b). One can see that the increase
of the distance hardly affects the paired PHs [see the first image in (a)
and that in (b)]. In addition, we have further simulated a scatterer of
single cylinder with unequal lateral distances to two borders of calcu­
lation window as shown in Fig. 2(c). As a result, a PNJ without curvature
is generated. This means that the curvature of beam is not caused by the
difference of the unequal distances to the two borders of the calculation
window. We may therefore conclude that the curvature of pH is not
caused by the effect of boundary conditions on the side border of scat­
terer. Instead, the formation of PHs can be ascribed to the optical in­
homogeneity of dual-dielectric structure. Regarding the left cylinder,
the medium on its left is air while the medium on its right is a mixture of
PDMS cylinder and air. The average refractive index of right medium is
larger than that of left medium. This optical inhomogeneity leads to
different phase delays of propagating waves on the two sides of scat­
terer, and hence causes the broken symmetry of wavefront phase dis­
tribution. Eventually, the interferences of these waves with different
wavefront deformations result in the formation of a curved beam. For
the right cylinder, there emerges a right PH induced by a similar optical
inhomogeneity to the case of left cylinder. In Fig. 2(a), one can see that
PHs bend less with the increasing spacing. Fig. 2(d) gives the quantita­
tive variational relations of θ and D. The θ decreases dramatically with
increasing D and reaches the maximum value 44.3◦ at D = 0.004λ,
consistent with the qualitative results in Fig. 2(a). The decrease of θ can
be attributed to the decrease of optical inhomogeneity of media on the
two sides of left or right cylinder. Still take the left cylinder as an
example. As D is increased, the contribution of high refractive index of
right cylinder to the average refractive index at the position of left cyl­
inder is gradually diluted by increasing distance between two cylinders,
while the left medium of left cylinder still has a fixed refractive index of
1. Thus, at the position of left cylinder, the difference of average
refractive index between the left and right media of left cylinder, i.e., the
optical inhomogeneity, decreases and results in the decreasing curvature
of PH.
It should be pointed out that, limited by our computer memory, the
simulation on the dual-structure with a small spacing of D = 0.004λ was
done using meshes with size of λ/50. We note that the mesh size in this
case is 5 times larger than the D value, resulting in that the numerical
results are not so accurate and only used for reference. Nevertheless, one
can see from Fig. 2(b) that the θ value in the case of D = 0.004λ follows
well the general tendency of the θ-D plot. In addition, the morphology of
the PH obtained in the case of D = 0.004λ seems to be sound.
Continue to discuss Fig. 2(e), it intuitively displays the light intensity
distributions along x-axis of paired PHs in the case of D = 0.004λ. These
distribution curves marked as A, B, C, …, G are respectively extracted at
points where the distances from focus point along y-axis increase from
0 to 2λ in steps of 0.4λ. The curve A is extracted at the focus point. The
vertex of each curve is marked in red, and two polylines connected by
these red dots indicate the propagation trajectories of a pair of sym­
metrical PHs. It is verified again that the paired PHs travel along two
symmetrical curved lines.
Based on that the curvature of PH is induced by optical in­
homogeneity, it is evident that besides changing spacing, the simulta­
neous change of refractive index or size of two same scatterer elements
in dual-dielectric structure can also change the optical inhomogeneity,
and hence change the bending angles of two symmetrical PHs. These
works are similar to the above investigation on the dependence of
bending angles on D and will not be repeated here.
3.2. Effect of scatterer size
Next, we study the PHs produced by a dual-dielectric structure
Fig. 2. Contours of light intensity of PHs generated by (a) dual-dielectric structure with varied spacing D, (b) the dual-dielectric structure with D = 0.004λ and
increased distance between the side of scatterer and the border of simulation window, (c) single cylinder with unequal lateral distances to two borders of simulation
window. (d) D dependence of bending angle θ of two PHs; (e) Intensity distribution curves along x-axis for PHs generated by the structure with D = 0.004λ, and the
curves marked A, B, …, D were extracted at different y positions.
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Optics and Laser Technology 157 (2023) 108673
consisted of two different scatterers. The difference of size or refractive
index between scatterer units introduces additional optical in­
homogeneity and further leads to the generation of asymmetrical PHs.
Firstly, it was carried out on a dual-dielectric structure in the case that
Hright varied in the range of 0.3λ-1.5λ while other parameters fixed at
initial values. From the morphology in Fig. 3(a), there emerges a pair of
asymmetrical PHs behind the structure. The increasing Hright affects the
light intensity distributions of both two PHs. This is distinct from the
case of single scatterer, in which the PNJ generated is only affected by
the structure itself. Fig. 3(c) shows more clearly the Hright dependence of
θleft and θright. The blue or red curve represents the relationship between
Hright and θleft or θright. As Hright is increased from 0.3λ to1.5λ, both
bending angles increase firstly and decrease then, and reach the
maximum value 42.7◦ at Hright = λ. One can see that θright varies more
than θleft. This is because the right PH is directly generated by the varied
cylinder, while the left PH is just indirectly affected by it.
It is worthwhile to point out that, although the light intensity dis­
tribution of PHs displays discontinuity as shown in Fig. 3(a) and (b), the
discontinuity does not obstruct us to evaluate the FWHM of the PH ac­
cording to their definition given in section 2. The detailed Hright
dependence of other relevant parameters of FL, FWHMx(z), and Imax/I0
are given in Fig. 3(e). Similarly, blue and red dotted lines separately
represent the cases of left and right PHs. As Hright is increased from 0.3λ
to1.5λ, the Imax/I0 of right PH increases from 3 to 12. The elongated
scatterer increases the optical path of lights propagating through scat­
terer, and hence renders these lights focus into a more compact focused
spot, quantitatively shown as an increasing intensity and decreasing
FWHMx(z) of right PH. When Hright < Hleft, the right PH has a larger
FWHMx(z) and lower Imax/I0. Instead, when Hright > Hleft, the case is just
opposite. When Hright = Hleft = λ, the dual-dielectric structure de­
teriorates to the symmetrical one in subsection 3.1 and generates two
symmetrical PHs with equal parameters values, shown as the in­
tersections of bule and red curves in Fig. 3(e). For left PH, Imax/I0 varies
little with Hright and the variation relationship between them is shown as
a quite flat curve with a slight drop. As the total energy of dual-dielectric
structure system is constant, the increase of light intensity of right PH
results in the decrease of light intensity of left PH. Similar to Imax/I0, as
Hright is increased, the FWHMx(z) of left PH also has a slight change due
to the indirect influence of elongated right cylinder. As for FL, the
variation of FL with Hright shows complex oscillation characteristics with
a first decrease and a next increase trend. In the case of single PNJ, the
FL decreases with the increasing length of scatterer in propagating di­
rection. As for PH, the curvature of light beam additionally causes the
decrease of FL, and the varying curvature (bending angle) leads to
different shortening of FL. Both the variational relations of θ to Hright and
FL to θ are not simple linear relationships. Thus, the relationship be­
tween FL and Hright is inconclusive.
After discussing the effect of height difference between two scatterer
units, we next focus on the effect of radius difference on the character­
istics of PHs. The dual-dielectric structure with a varying Rright generates
a pair of asymmetrical PHs as shown in Fig. 3(b). As Rright is increased,
the left PH becomes shorter and right PH becomes longer and intensive.
Fig. 3(d) exhibits detailed Rright dependences of θleft and θright, indicated
by blue and red polylines. As Rright is increased from 0.4λ to1.4λ, both
θleft and θright increase and then decrease, reaching the maximum value
42.7◦ at Rright = λ. The Rright dependences of other parameters are also
exhibited in Fig. 3(f). As Rright is increased from 0.4λ to1.4λ, the Imax/I0,
FWHMx(z) and FL all increase. The increase of Imax/I0 can be explained
from the viewpoint of ray optics. The formation of PNJ is similar to the
focusing procedure of a lens. The scatterer can be regarded as a lens and
the PNJ can be regarded as a focused point. The cylinder with a larger
radius corresponds to the lens with a larger numerical aperture, which
has a stronger ability of collecting light. The PH is a variant of PNJ and
has a similar focusing procedure to PNJ. Thus, the right cylinder with
increasing Rright generates a PH with larger Imax/I0, and a more intensive
right PH leads to a weaker left PH. The increase of right FWHMx(z) is
related to the geometry features of the widened right scatterer. The
smaller the radius of scatterer is, the more compact the focused point is.
Thus, a larger Rright causes a widened focused spot with an enlarged
FWHMx(z). As for left PH, the Imax/I0 decreases slightly. The FWHM and
FL change little.
The investigation on the effect of size difference shows that both of
the bending angles of asymmetrical PHs can be easily changed by
changing the length or radius difference between the scatterer units of
dual–dielectric structure. Other characteristics of two PHs can also be
tailored by the changed size of either one scatterer. And those param­
eters of the PH generated by the varied scatterer change more. This is
because this PH is directly produced and controlled by the varied scat­
terer, while the other PH generated by the fixed scatterer is just indi­
rectly affected by the varied one.
3.3. Effect of refractive index difference
In this subsection, the investigation is focused on the optical in­
homogeneity introduced by refractive index difference between scat­
terer units in dual–dielectric structure. It was carried out in the case that
the right refractive index ratio nright between the refractive indices of
right cylinder and surrounding medium varied in the range of 1.2–1.8
while other parameters fixed at R = H = λ, nleft = 1.4, and D = 0.1λ. As
shown in Fig. 4(a), a pair of asymmetrical PHs are generated. As nright is
increased from 1.2 to 1.8, the right PH has an increasing light intensity
and elongating focal length. This is similar to the case of single PNJ, in
which the scatterer having a higher refractive index generates a PNJ
with larger intensity, narrower width and shorter FL. The nright de­
pendences of the characteristic parameters of two PHs are quantitatively
described by the blue and red curves in Fig. 4(b) and (c). In Fig. 4(b), it is
clear that θleft and θright have unequal values unless in the case of nleft =
nright = 1.4. Both two bending angles increase and then decrease. The
θleft reaches the maximum value 43.9◦ at nright = 1.5, and the θright rea­
ches the maximum value 47.5◦ at nright = 1.6. The variations of other
parameters in Fig. 4(c) have the following features. As nright is increased,
the right PH has an increasing Imax/I0, and the left PH has an Imax/I0 with
less variation. When nleft > nright, the left PH has a larger intensity,
smaller FWHM, and shorter FL. When nleft < nright, the case is just
opposite.
3.4. Effect of scatterer shape
Next, we study the dual–dielectric structure consisted of two scat­
terers of other shape, including cuboids, spheres, and unflipped/flipped
cylinders. For comparation, these paired scatterers still adopt the initial
parameters of refractive index n = 1.4 and the length along y-axis of Ly
= λ. The cuboids have side lengths along x- and z-axes of Lx(z) = 2λ, and
the spheres and cylinders have radii of λ. Fig. 5(a)-(d) gives the light
intensity distributions of paired PHs generated by these dual-structures
in the case of D = 0.1λ, 0.5λ, and λ. One can see that each structure
generates a pair of symmetrical PHs bent outward. The contours of light
intensity of PHs produced by cuboids and unflipped cylinders are
similar, both of which are features by a bar-shaped light spot emerging
from the middle of two paired PHs. The bar-shaped spot generated by
cuboids is more evident as shown in Fig. 5(a). Due to that the cuboid and
unflipped cylinder have same rectangular projections onto xy plane, the
intensity difference between two bar-shaped spots is attributed to the
structural difference in the z direction. The cuboid or unflipped cylinder
separately has a square or circle projection onto xz plane. The cuboid has
four straight sides without curvature, while the circle has the curved
edge with greater curvature and hence a betterer refractive ability. Thus,
the structure consisted of the unflipped cylinders generates a pair of
compact PHs with less stray lights, resulting in a weaker interference
effect of these lights and a less obvious bar-shaped spot [see in Fig. 5(c)].
Instead, the spheres and flipped cylinders generate paired PHs without
bar-shaped spot as shown in Fig. 5(b) and (d). This is because both of
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Optics and Laser Technology 157 (2023) 108673
Fig. 3. Contours of light intensity of PHs generated by dual-dielectric structure with varied (a) Hright, and (b) Rright; the variation of (c) θ with Hright, and (d) θ with
Rright; (e) Hright and (f) Rright dependences of characteristic parameters of two PHs.
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Optics and Laser Technology 157 (2023) 108673
Fig. 4. (a) Contours of light intensity of PHs generated by the dual-dielectric structure with the fixed D = 0.1λ, nleft = 1.4, and varied right refractive index ratio
nright; Right refractive index ratio nright dependences of (b) θ, and (c) Imax/I0, FWHMx, FWHMz and FL of two PHs.
Fig. 5. Contours of light intensity of PHs generated by the dual-dielectric structure of 2 (a) cuboids, (b) spheres, (c) unflipped cylinders and (d) flipped cylinders with
the lengths along z-axis similar to the lengths along x- and y-axes, and (e) 2 cuboids and 2 flipped cylinder with the lengths along z-axis 20 times larger than the
lengths along x- and y-axes.
them have circular projections onto xy plane with good focusing ability.
Moreover, compared with the flipped cylinders, two spheres generate
paired PHs with less stray lights due to that their 3D spherical symmetric
structures have greater curvature in all directions. Besides, the charac­
teristic parameters of paired PHs in Fig. 5 also vary with the spacing D.
In the above discussion, we have demonstrated that the array of
cuboids can generate a more intensive bar-shaped spot between paired
PHs due to its straight side with worse refractive ability. The formation
of this bar-shaped spot can be regarded as a superposition result of the
incident lights and scattered lights from xy planes at different z posi­
tions. Next, we further study the enhanced superposition effect by
elongating scatterers along z-axis. The first image in Fig. 5(e) exhibits
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Optics and Laser Technology 157 (2023) 108673
the intensity distributions of light spots scattered by cuboids array with
large Lz [the side length along z-axis Lz is 20 times larger than the side
lengths along x- and y-axes Lx and Ly (also the wavelength λ)]. One can
see that besides a pair of weak PHs, there emerges an intensive PNJ with
long FL behind them. This is another phenomenon and the further study
on Lz dependence of relevant parameters of this large PNJ would be the
future work.
To summarize, the dual-dielectric structure consisted of cuboids with
large Lz (Lz > 20Lx and Lz > 20Ly) can generate a pair of weak PHs and
one intensive PNJ, while the structure with small Lz, i. e., the side length
along z-axis is similar to the side lengths in x- and y-axes (Lz ~ Lx and Lz
~ Ly), only generates a pair of symmetrical PHs. It gives a hint that for
the multi-dielectric structure consisted of cuboids, the length along zaxis Lz plays a significant role in determining the features of focusing
pattern. If there is a large Lz, the superposition effects of the incident and
scattered lights from xy planes at different z positions are enhanced,
resulting in that the bar-shaped spot is enhanced and transforms into an
intensive PNJ. Regarding the structure consisted of two flipped cylin­
ders, the bar-shaped spot can also be enhanced and transforms into an
intensive PNJ by increasing the length along z-axis as shown in Fig. 5(e).
With regard to unflipped cylinders, they have circular projections onto
xz plane with the same length along x- and z-axes. And the length of this
structure along z-axis would never be considerably larger than the
length along x-axis. Thus, it is impossible to transform the bar-shaped
spot generated by the structure consisted of two unflipped cylinders
into a large PNJ by increasing the length along z-axis. The same re­
striction also exists in the case of spheres array.
3.5. Effect of scatterer number
With regard to the multi-dielectric structure of odd scatterer units,
we studied the tri-dielectric structure as a representative. Due to the
good scalability of multi-dielectric structure, a tri-dielectric structure
can be obtained by simply adding one scatterer unit to the dualdielectric structure. Compared with the dual–dielectric structure, the
tri‑dielectric one with more elements has more freedom to tailor the
performance of PHs array. For example, there are two adjacent spacings
in tri‑dielectric structure defined as Dleft and Dright, and the structures
with equal or unequal Dleft and Dright generates the PHs array with
different features. Firstly, the structure with equal Dleft and Dright is
studied in the range of 0.004λ-5λ. As a representative, the first image in
Fig. 6(a) gives the contours of light intensity of PHs generated in the case
of Dleft = Dright = 0.2λ. Regarding the middle scatterer of this structure,
the media on its both sides are symmetrical; but regarding the bilateral
scatterers, similar to the case of dual-dielectric structure, the media on
their both sides are asymmetrical. As a result, the middle unit produces a
PNJ, while two lateral scatterers generate a pair of symmetrical PHs bent
outward. Secondly, the structure with unequal Dleft and Dright is also
studied. The last two images in Fig. 6(a) display the light intensity dis­
tributions of PHs produced by the tri-dielectric structure with (Dleft,
Dright) values of (0.2λ, 0.8λ) and (0.4λ, 0.6λ). Distinct from the case of
Dleft = Dright, the structures with Dleft ∕
= Dright produce three asymmet­
rical PHs with different bending degrees. This is because the difference
between Dleft and Dright breaks the media symmetry on the both sides of
middle scatterer, resulting in the generation of the middle PH.
In addition, the effects of size difference between scatterer units on
the features of PHs array were also investigated. It was carried out on the
basis of a tri–dielectric structure in the case that the height or radius of
the middle cylinder Hmid or Rmid varied in the range of 0.36λ-1.2λ while
other parameters fixed at the initial values at R = H = λ, n = 1.4, Dleft =
Dright = 0.5λ. Fig. 6(b) gives the light intensity distributions of PHs
generated by the structure with Rmid increasing from 0.4λ to 1.4λ. One
can see that each structure generates a pair of symmetrical PHs and one
PNJ. The intensity and FL of the middle PNJ increase due to the
increased numerical aperture caused by increasing Rmid. The bending
angles θ of two PHs vary with Hmid and reach the maximum value 44.4◦
Fig. 6. Contours of light intensity of PHs generated by the tri-dielectric struc­
ture with (a) varied D, (b) varied Rmid, and (c) varied Hmid. (d) Rmid (red line)
and Hmid (blue line) dependences of θ.
at Rmid = 0.6λ. Based on this, we studied the dependence of Hmid on θ in
the case of Rmid = 0.6λ. From the morphologies in Fig. 6(c), it is notable
that all the middle PNJs generated by the structure with Rmid = 0.6λ
cannot shoot out. As Hmid is increased from 0.36λ to1.2λ, the middle PNJ
has an increasing intensity. Fig. 6(d) gives the detailed quantitative re­
lationships between θ and Rmid (Hmid). The θ increases and then de­
creases, reaching the maximum value 44.4◦ when Hmid = λ and Rmid =
0.6λ.
Based on that the multi-dielectric structure has a large flexibility in
structural adjustment, it is easy to tailor the performance of PHs array by
increasing or reducing the scatterer number. In the case of R = H = λ, n
= 1.4 and D = 0.5λ, the contours of light intensity of PHs array gener­
ated by a multi-dielectric structure of N scatterer units are exhibited in
Fig. 7(a). The structure of odd scatterer units generates one middle PNJ
and (N-1)/2 pairs of symmetrical PHs. Particularly, single cylinder
produces one PNJ as expected. As for these (N-1)/2 pairs of PHs, each
pair of symmetrical PHs are produced by a pair of symmetrically posi­
tioned scatterer units of the structure. Paired PHs usually bend outward,
7
Y.-J. Yang et al.
Optics and Laser Technology 157 (2023) 108673
Fig. 7. Contours of light intensity of PHs array generated by the multi-structure of (a) cylinders with the scatterer number of N = 1, 2, 3, 4, 5, 6, 7; (b) seven cylinders
with the spacing from left to center (center to right) of D = 0.8λ, 0.6λ and 0.4λ (the first image); six cylinders formed by removing the middle unit of the structure in
the first image (the second image).
and the ones located at the outermost have larger bending angles. The
middle PNJ has the smallest bending angle of 0. These features are
associated with the formation of PHs array. The position at the outer­
most has the greatest optical inhomogeneity of the media on its both
sides, resulting in the PH generated by the scatterer located at the
outermost has the largest curvature. The structure of even scatterer units
generates N/2 pairs of PHs. And the paired PHs located at the outside
bend more. The increase of scatterer number introduces more elements
into the system of multi-dielectric structure and improves the flexibility
of tailoring the performance of PHs array generated. With a large flex­
ibility in structural adjustment, the features of PHs array can also be
adjusted by appropriately changing the spacing between two adjacent
scatterers. Two images in Fig. 7(b) separately exhibit the PHs arrays
generated by the multi-dielectric structures of seven and six scatterer
units with properly designed geometry. In the first image, the spacings D
between every-two adjacent units from left to center and center to right
are 0.8λ, 0.6λ and 0.4λ in turn, and the middle cylinder unit has equal
radius and height of 0.6λ. In the second image, the multi-dielectric
structure of six scatterer units is formed by removing the middle unit
of the structure shown in the first image. Distinct from the PHs bent
outward in Fig. 7(a), the paired PHs located at the innermost bend in­
ward as shown in Fig. 7(b). It means that the characteristic parameters of
PHs array, even the bending direction, can be effectively changed by an
adequately designed geometry of the multi-dielectric structure. By
immediately adjusting the spacing and scatterer number of the struc­
ture, it is possible to dynamically control the characteristic parameters
of PHs array in real time. The narrowest gap between two PHs bent
inward is similar to 1 μm. Thus, this paired PHs bent inward can be used
as a pair of optical pliers to manipulate a biological cell or other microscale particle. By changing the geometrical parameters of the multidielectric structure, the opening angle of the optical pliers can be
easily adjusted to realize the manipulation of a cell or a particle with a
varied size. It should be pointed out that the light localization of the
PNJ/PH in subwavelength scale enables to reduce the light intensity
required to trap a nano-/micro-object. In 2015, a PNJ with a focal in­
tensity 4 times the intensity of incident light has been successfully used
to track and trap a 50-nm-sized particle [6]. One year later, researchers
reported a lower PNJ intensity ratio of 3 used for trapping and operating
of a 100-nm-sized particle [37,38]. Since our PHs array has an intensity
ratio>6, it can be also used for the manipulation of particle.
Whether the array of PHs bends outward or inward, it shows great
potential for the applications in some special scenarios. The paired PHs
bent outward may find applications in the fields of controlling on/off
state of optical switches in micro-/nano-scale small space, simulta­
neously sensing and tracking multiple particles, etching and machining
special patterns array, and connecting tiny multiple electrodes on the
photonic chip [16], while those paired PHs bent inward may find ap­
plications in the fields of batch manipulation on biological cells or
nanoparticles [12–13,15]. These above-mentioned application pros­
pects of PHs array will be confirmed by further experiments in the future
work.
Finally, it is worthwhile to point out that the above-mentioned
studies concern only 1D arrays of scatterers. One may ask how about
the results in the case of 2D configurations. To make the argument clear,
we have simulated a multi-dielectric structure consisted of four cylin­
ders symmetrically arranged in a square with a side length of 2.5λ. Fig. 8
(a) shows the contour of light intensity of PHs projected onto xy or yzplane. Fig. 8(b) shows the case of the projection onto the plane defined
by the square’s diagonal and normal lines. One can see that each PH
bends outward. It appears that and the bending of the PHs in Fig. 8(b) is
more evident than that in Fig. 8(a). This is because the bending occurs in
the plane defined by the square’s diagonal and normal lines. The feature
is further verified by distribution of Poynting vector. Fig. 8(c) shows 3D
Poynting vector distribution of four PHs generated by the above8
Y.-J. Yang et al.
Optics and Laser Technology 157 (2023) 108673
Fig. 8. Contours of light intensity of PHs generated by a multi-dielectric structure consisted of four cylinders symmetrically arranged in a square with a side length of
2.5λ and projected onto (a) xy or yz plane, (b) the plane defined by the square’s diagonal and normal lines. (c) 3D distribution of Poynting vector of four
PHs generated.
mentioned 2D cylinder square array, which is depicted on the bottom of
Fig. 8(c) for convenience. The color of Poynting vector reflects light
intensity and the arrow indicates the direction of power flow. The
feature that the PHs bend outward is also discernible in Fig. 8(c). In a
word, similar to the case of 1D arrayed multi-dielectric structure, 2D
configuration can also generate the PHs bent outward. Next, we
demonstrate briefly that the PHs bent inward can be also produced by a
properly designed 2D configuration. Here, we demonstrate it by refer­
encing the scheme of 1D configuration. On the basis of the 1D arrayed
multi-dielectric structure [Fig. 7(b)], we have designed a 2D configu­
ration used to generate the PHs bent inward. The 2D configuration is
shown at the bottom of Fig. 9(a). A pound-shape (#) 2D architecture is
proposed. The four lines of the 2D pound array have the same geometry.
Each line includes six scatterer units with the same arrangement as that
in Fig. 7(b). This proves that the proposed multi-dielectric structure has
great scalability. The simulated distribution of 3D Poynting vector of
PHs generated by the pound-shape 2D structure is shown in Fig. 9(a).
For clarity, only the central four ones in the PHs array are selected and
shown. We note that the four PHs reveal a tendency of inward bending.
Similar to the outward bending case, the inward bending takes place
also in the plane defined by the square’s diagonal and normal lines. To
clarify the argument, in Fig. 9(b) we show the contour of light intensity
of PHs projected onto xy or yz-plane and in Fig. 9(c) we show the case of
the projection onto the plane defined by the square’s diagonal and
normal lines. One can see that the projected PHs in Fig. 9(c) bend more
obviously than those in Fig. 9(b). As operating arms, the four PHs bent
outward form an optical anchor to manipulate a ring shape micro-object
for example, and the four ones bent inward form an optical gripper to
grasp a biological cell or particle.
4. Conclusion
The study allows us to reach the following conclusive results.
(1) An array of PHs can be generated by the multi-dielectric struc­
ture. Its formation can be explained in terms of the interference of
scattered lights having different wavefront deformations induced
by optical inhomogeneity.
(2) A study on effect of spacing between two adjacent scatterers
shows that paired symmetrical PHs are generated in the studied
spacing range of 0.004λ-5λ. Their bending angles decrease with
the increased spacing because of the decreased optical
inhomogeneity.
(3) A study on effect of size difference between scatterer units shows
that a dual-dielectric structure consisted of two cylinders with
different lateral/longitudinal dimensions generates two asym­
metrical PHs with different bending angles. It is ascribed to
different optical inhomogeneities at the two cylinder units. The
bending angles can be tailored by changing one cylinder
geometry.
(4) A study on a dual-dielectric structure with nleft ∕
= nright shows that
the performances of two asymmetrical PHs can be effectively
tailored by refractive index difference between two scatterers.
Their bending angles can be tailored by changing refractive index
of one scatterer unit.
(5) A study on dual-dielectric structure with different scatterer
shapes shows that the scattering characteristics of array of
unflipped (flipped) cylinders are similar to those of array of
Fig. 9. (a) 3D distribution of Poynting vector of PHs generated by the proposed
2D pound-shape configuration. Contour of light intensity of PHs projected onto
(b) xy or yz-plane, and (c) the plane defined by the bottom square’s diagonal
and normal lines.
9
Y.-J. Yang et al.
Optics and Laser Technology 157 (2023) 108673
cuboids (spheres). This is associated with geometrical charac­
teristic of the structure projected onto xy plane. In specific, the
structure of unflipped cylinders or cuboids with long length in zdirection generates a long and intensive PNJ located behind two
small and weak paired PHs.
(6) A study on tri-dielectric structure shows that, as Dleft = Dright,
Hmid = λ and Rmid = 0.6λ, two lateral PHs display the largest
bending angles of 44.4◦ . The geometrical and optical parameters
of any scatterer unit can be used to tailor the performance of the
PHs array.
(7) Increasing number of scatterer units improves the flexibility to
tailor the performance of PHs. The PHs array generated by a
structure of odd or even scatterer units exhibits quite different
characteristics. Paired PHs usually bend outward, and the ones
located at the outermost have larger bending angles. Paired PHs
bent inward can also be generated by a multi-dielectric structure
with a special designed geometry. We show that 2D PHs array
having features similar to the case of 1D PHs array can be ob­
tained by extending the 1D configuration to a properly designed
2D configuration. The 2D arrayed PHs bent outward/inward form
an optical anchor/gripper that may find its application in the
field of 3D manipulation.
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Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Data availability
No data was used for the research described in the article.
Acknowledgement
This work is supported by the National Natural Science Foundation
of China under Project no. 61875148.
Author agreement
All of authors agree the submission, and would like to state that: the
article is original, has been written by the stated authors who are all
aware of its content and approve its submission, has not been published
previously, it is not under consideration for publication elsewhere, no
conflict of interest exists, the article will not be published elsewhere in
the same form, in any language, without the written consent of the
publisher.
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