Optical Forces in Metal/Dielectric/Metal Fishnet Metamaterials in the Visible Wavelength Regime

Abstract

The optical force is calculated on a nanoparticle in close proximity to the surface of a fishnet metamaterial based on a metal/dielectric/metal film when illuminated at visible wavelengths. We show that the optical force can be enhanced by the strong magnetic dipole in the fishnet metamaterial. In contrast to other plasmonic nanostructures, which exhibit an attractive force in all regions, our presented structure provides good flexibility in pushing and dragging particular size nanoscale particles at certain distances above the surface of the structure. Therefore, it is suitable for size selection and optical trapping of nanoscale particles at illumination intensities of (1 mW/μm2). At this power level, calculation shows that the optical force can be up to four orders of magnitude larger than the gravitational force for a 100-nm radius nanoparticle.

Full text

Optical Forces in Metal/Dielectric/Meta Fishnet
Metamaterials in the Visible Wavelength Regime
Volume 4, Number 5, October 2012
Tun Cao
Lei Zhang
Martin J. Cryan
DOI: 10.1109/JPHOT.2012.2218589
1943-0655/$31.00 ©2012 IEEE

Optical Forces in Metal/Dielectric/Metal
Fishnet Metamaterials in the Visible
Wavelength Regime
Tun Cao,1Lei Zhang,1and Martin J. Cryan2
1Department of Biomedical Engineering, Faculty of Electronic information and Electrical Engineering,
Dalian University of Technology, Dalian 116024, China
2Department of Electrical and Electronic Engineering, University of Bristol, Bristol, BS8 1TR, U.K.
DOI: 10.1109/JPHOT.2012.2218589
1943-0655/$31.00 Ó2012 IEEE
Manuscript received August 8, 2012; revised September 7, 2012; accepted September 9, 2012. Date of
current version September 25, 2012. This work was supported by the National Natural Science
Foundation of China under Grant 61172059 and 61071123, by the Ph.D. Programs Foundation of the
Ministry of Education of China under Grant 20110041120015, by the Postdoctoral Gathering Project of
Liaoning Province under Grant 2011921008, and by the Fundamental Research for the Central
University under Grant DUT12JB01. Corresponding author: T. Cao (e-mail: caotun1806@dlut.edu.cn).
Abstract: The optical force is calculated on a nanoparticle in close proximity to the surface
of a fishnet metamaterial based on a metal/dielectric/metal film when illuminated at visible
wavelengths. We show that the optical force can be enhanced by the strong magnetic dipole
in the fishnet metamaterial. In contrast to other plasmonic nanostructures, which exhibit an
attractive force in all regions, our presented structure provides good flexibility in pushing and
dragging particular size nanoscale particles at certain distances above the surface of the
structure. Therefore, it is suitable for size selection and optical trapping of nanoscale
particles at illumination intensities of ð1mW=m2Þ. At this power level, calculation shows
that the optical force can be up to four orders of magnitude larger than the gravitational force
for a 100-nm radius nanoparticle.
Index Terms: Metamaterials, optical force, nanohole arrays.
1. Introduction
The momentum exchange between electromagnetic radiation and matter is a fundamental process
that has been exploited for remarkable applications such as laser cooling [1]–[3] and optical
tweezers [4], [5]. In particular, optical tweezers have been applied to trap and manipulate a wide
range of biological and nonbiological objects including viruses, cells, colloidal spheres, and
intracellular organelles. However, in conventional optical tweezers, focused light usually results in a
repelling force on the particles due to positive radiation pressure and momentum conservation [6]–[9].
In a recent study, researchers have investigated the optical forces produced by nonparaxial
gradientless beams and found that the forces can drag suitable particles all the way toward the light
source [10], [11]. However, optical manipulation by light beams suffers from the diffraction limit for the
trapping volume since the momentum transfers from the propagating fields to the particles [12], and
the exposure time of a trapped particle is limited due to the high optical power and focusing of the laser
beam [13]. In order to overcome these limitations and extend the range of in-plane optical
manipulation, surface plasmonic (SP) systems have been proposed to address the problem [14]–[16].
In SP systems, momentum transfer occurs from evanescent fields near the surface of metal/dielectric
interfaces rather than propagating fields, allowing for the possibility of nanometric control of the
particles [17]–[21]. However, the use of a prism in conventional SP techniques makes it difficult to
Vol. 4, No. 5, October 2012 Page 1861
IEEE Photonics Journal Optical Forces in MDM Fishnet Metamaterials

scale down the system [22] and limits its integration with micro and optofluidic chips, which is
important in transitioning biological techniques from the lab to lab-on-chip [23].
Recent efforts have turned toward the near-field confinement properties of plasmonic
nanostructures [24], [25]. Compared with conventional optical tweezers, it has become clear that
plasmonic nanostructures can provide good control over optical forces on nanoscale particles by
creating high local field enhancements [26]–[29]. Specifically, an array of Au bowtie nanoantennas
(BNAs) for optical trapping and manipulation of submicrometer sized objects has been proposed [30].
These periodic plasmonic structures have a strong electrical resonance because each unit cell
functions as an electrical quadrupole. It results in a large intensity, creating a very large optical
dragging force, which exceeds the electromagnetic radiation pressure, hence always pulling the
particles toward the surface. Namely, the optical force induced by the electrical dipole is always
dragging in all regions. To our knowledge, most plasmonic nanostructures are mainly focused on the
electrical dipolar force, and the optical manipulation on the nanoparticles using magnetic dipolar force
has been rarely reported.
Fishnet metamaterials based on metal/dielectric/metal (MDM) films offer the opportunity for giving
rise to a strong magnetic resonance. To date, much research on MDM fishnet metamaterials has
been focused on achieving double negative index materials [31]–[34]. In this paper, we demonstrate
the use of fishnet metamaterials for optical manipulation and size selection of nanoscale particles.
Compared with other plasmonic nanostructures exhibiting trapping force in all regions, we find that
the fishnet embedded in a MDM film provides a good flexibility in pushing and dragging certain
nanoscale particles at particular distances above the surface of the structure. This is due to the
interaction between the near-field electromagnetic force and electromagnetic radiation pressure. We
show that this characteristic is suitable for size selection and optical trapping of nanoscale particles
with and without loss at illumination intensities of ð1mW=m2Þin the visible region. We also show
that the optical force is sufficient to overcome gravity. In particular, we investigate the impact of the
geometry (round and elliptical holes) to show that structures formed from elliptical nanohole arrays
(ENAs) can provide a greater trapping force than structures based on round nanohole arrays (RNAs)
due to their stronger magnetic dipole resonance. Furthermore, our structure offers the potential for
direct integration into microfluidic platforms since it can excite surface plasmon resonance (SPR)
without a prism or grating structure [35]. The footprint of a fishnet array is small relative to that
typically required in reflective mode SPR, enabling miniaturization and integration into microfluidic
architectures and multiplexed analysis, and allowing higher spatial resolution [36].
2. Method
In this section, we use Maxwell’s stress tensor method to calculate the optical force on a dielectric
particle near the surface of the MDM fishnet metamaterial. We placed six plane monitors
surrounding the particle to obtain the electric- and magnetic-field components. The field
components are related with Maxwell’s stress tensor by the following formula:

$opt
ij ¼"EiE
jþHiH
j1
2ij "jEj2þjHj2
 (1)
where 
$opt
ij is the Maxwell stress tensor, Eiand Hicorrespond to the electric and magnetic fields,
"represents the electric permittivity and represents the magnetic permeability for the particle on
which optical force is applied, and ij is the Kronecker delta. The total optical force acting on the
particle has been calculated by the following formula [37], [38]:
ðFtotalÞ¼1
2Re Z
ZS
ðb
n
$optÞdS
 (2)
where Sis a bounding surface (six planes) around the nanoparticle, and b
nis the outward normal to
the surface of the six planes mentioned before. The total optical force Ftotal (assuming Casimir and
heating effects to be negligible) consists of electromagnetic radiation pressure Frad and the near-
field electromagnetic force Fnr induced by the MDM fishnet metamaterial, where Frad depends on
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the spectrum of the coefficients of the structure according to
Frad ¼ð2RþAÞP
c(3)
where Ris reflectance, Ais absorbance, Pis the power of the incident light, and cis the speed of
the light in vacuum [39]. All fields are well defined, and the integration can be straightforwardly
carried out with EMexplorer (a 3-D FDTD commercial software).
3. Results and Discussions
Our simulated structure is a sandwich of 30-nm-thick Au, 120-nm-thick Al2O3, and 30-nm-thick Au.
The proposed metamaterials are suspended in air and consist of a periodic arrangement of identical
meta-atoms. Such freestanding metamaterials can be experimentally realized by using undercutting
techniques [40], [41]. They are arranged on a square lattice with 350-nm period. The meta-atoms
are either round or elliptical holes, which penetrate through the sandwich structure, where the round
Fig. 1. (a) Schematic of a MDM fishnet metamaterials consisting of an Al2O3dielectric layer between
two Au films perforated with a square array of round holes suspended in air. The lattice constant is
L¼350 nm and hole diameters are d¼90 nm. A particle can be dragged from original position ato new
position a0or pushed from original position bto a new position b0. (b) Illustration of the square lattice of a
RNA. (c) Schematic of a MDM fishnet metamaterials consisting of elliptical holes suspended in air. The
lattice constant is L¼350 nm and hole diameters are d1¼110 nm and d2¼90 nm. (d) Illustration of the
square lattice of an ENA.
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hole diameter is d¼90 nm, and elliptical hole diameters are d1¼110 nm and d2¼90 nm.
Schematic representations of these structures are shown in Fig. 1. The gold permittivity is well
approximated by a Drude model, "ð!Þ¼1ð!2
p=½!ð!þi!cÞÞ, where !p¼1:37 1016 s1is the
plasma frequency and !c¼2:04 1014 s1is the collision frequency for bulk gold [42]. The
refractive index of Al2O3is 1.62. To achieve better accuracy of the plasmonic effect, we use a very
fine mesh size (2 nm) near the structure. Perfectly matched layer (PML) boundary conditions are
employed in the z-direction and periodic boundaries in the x–yplane. We study the structures in
ppolarization, where the electric field is parallel to the short axis of the elliptical holes as shown in
Fig. 1(c). Both of the structures are illuminated in the visible regime with a power density of
ð1mW=m2Þ. The power intensity is chosen to be readily tolerated by the thin Au films and to create
sufficient optical force to exceed the gravitational force [30].
Fig. 2(a) presents the transmittance, reflectance, and absorbance of the RNA. It shows that
transmittance is very low and a reflectance dip and absorbance peak are observed around 763 nm
where AbsorbanceðAÞ¼1ReflectanceðRÞTransmittanceðTÞ. These are most likely deter-
mined by the magnetic resonance, which results from a pair of finite-width Au films separated by a
dielectric layer along the direction of the incident light [43]. Using the procedure described in
Section 2, we then calculated Frad,Frad , and Ftotal on a dielectric particle, which is 160 nm above the
surface of the structure and is shown in Fig. 2(c). We assume that the particle has a relative
dielectric permittivity "p¼2:25 and the radius of the particle Rp¼100 nm It can be seen that Ftotal,
Fnr, and Frad are resonant. The dispersion of Frad is decided by absorbance and reflectance of the
RNA and has a local minimum at a wavelength of 763 nm corresponding to the absorbance peak
shown in Fig. 2(a).
In Fig. 2(b), the absorbance at 775 nm with the ENA is higher than for the RNA shown in Fig. 2(a).
This is due to better impedance matching between air and the metamaterial [33]. Fig. 2(d) shows
that Ftotal in the ENA at 775 nm is stronger than the RNA near 763 nm shown in Fig. 2(c). It is
Fig. 2. Reflectance (R), transmittance (T), and absorbance (A) of the MDM fishnet metamaterials
for (a) the RNA and (b) the ENA; Frad ,Fnr, and Ftotal on the nanoparticle (Rp¼100 nm and "p¼2:25),
which is 160 nm above the MDM fishnet metamaterials when illuminated under an incident field of 1
mW=m2for (c) the RNA and (d) the ENA.
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believed that the larger negative value of Ftotal in the ENA is a consequence of the structure’s higher
absorbance induced by the stronger magnetic dipole resonance.
It is important to note that Ftotal is attractive for both ENA and RNA, creating a trap at 160 nm
above the MDM fishnet metamaterials that can be used for biosensing. In particular, Fig. 3(a) and (b)
show that the resonance wavelengths of Reðeff Þare close to the resonance wavelengths of Ftotal
where the dielectric particle can obtain the highest attractive force. The calculated effective
permeability eff and effective permittivity "eff shown in Fig. 3(b) and (c) can be extracted from the
effective refractive index neff and impedance eff by the formulas eff ¼neff eff and eff ¼neff=eff ,
where the calculation of neff has been discussed in [43]. As shown in Fig. 3(b), the real part of the
permeability ReðeffÞhas a resonant modulation around 775 nm for the ENA and 763 nm for the RNA,
where Reðeff Þfor the ENA has a more negative value than the RNA due to a better impedance match
between the metamaterial and the surrounding claddings [33]. It indicates that the value of Ftotal is
associated with the strength of the magnetic dipole resonance.
Further simulations are performed to study the relationship between the optical force and
magnetic dipole resonance, we investigate total magnetic-field intensity distributions of all the
components HxHyHzfor the magnetic resonant wavelengths 775 nm for the ENA and 763 nm for
the RNA at a plane shown in Fig. 1. The results are calculated for normal incidence. In Fig. 4, the
arrows represent the electric displacement current, whereas the color represents the magnitude of
the magnetic field. A loop in electric displacement current, which leads to a local magnetic-dipole
moment, can be observed [43]. Fig. 4 also shows that the total magnetic-field intensity can be
efficiently confined between the two Au layers for both of the structures. Therefore, these structures
support a magnetic resonance at which light is trapped and strongly absorbed to create a higher
value of the optical force [44]. Specifically, the localized magnetic fields in the ENA shown in
Fig. 4(a) are enhanced more strongly than for the RNA shown in Fig. 4(b), implying that a greater
magnetic dipolar force can be obtained. Calculation shows that Ftotal and Frad under 1-mW=m2
Fig. 3. (a) Comparison of Ftotal on the nanoparticles (Rp¼100 nm and "p¼2:25), 160 nm above the
MDM fishnet metamaterials for RNA and ENA when illuminated under an incident field of 1 mW=m2;
(b) Comparison of the real part of the permeability ðReðeffÞÞ between the ENA and RNA; (c) Comparison
of the real part of the permittivity ðReðeff ÞÞ between the ENA and RNA; (d) Comparison of the real part of
the refractive index ðReðeff ÞÞ between the ENA and RNA.
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illumination intensity is four orders larger than the gravitational force for the 100-nm radius
nanoparticle (0.04 fN). Thus, we only consider the optical force exerted on the nanoparticle. It is
also found that the magnetic dipolar force depends on wavelength therefore providing dynamic
controllability and spectral selectivity.
We now study the possibility of using MDM fishnet metamaterials to produce total optical forces
Ftotal with different directions, to drag and push away particles at certain distances. In Fig. 5, we take
an ENA as an example and first compute Ftotal at a fixed magnetic resonant wavelength of 775 nm
for nanoparticles with Rp¼60 nm;80 nm;100 nm;and 120 nm along the z-axis; all particles have
a relative dielectric permittivity "p¼2:25. It shows that Ftotal has different signs along the z-axis due
to the interaction between Fnr and Frad. When the particle is close to the surface of ENA, the value of
Ftotal is negative. This is because the total magnetic-field intensity is locally enhanced near the
surface of the ENA, which results in a high value of the attractive near-field electromagnetic force
Fnr, which is dominant and drags the nanoparticle toward the surface of the structure. When the
Fig. 5. (a) Ftotal exerted on the nanoparticle along the zaxis above the ENA with different radius Rpat a
fixed magnetic resonant wavelength 775 nm; (b) zoom in picture of (a) around trapping positions.
Fig. 4. Map of the normalized total magnetic-field intensity distribution and electric displacement
current at the resonance wavelength for a cross section along the plane (z–yplane) for (a) the ENA
at 775 nm; (b) the RNA at 763 nm.
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particle moves to a certain distance above the surface, the reduced total magnetic-field intensity
gives a smaller value of Fnr, the repelling radiation pressure Frad will overcome Fnr to push the
particle upwards, and Ftotal changes to become positive. The trapping positions are defined as the
place where Ftotal changes sign [13]. In Fig. 6, we identify four such points near Z¼250 nm for
Rp¼60 nm;80 nm;100 nm;and 120 nm correspondingly. It also indicates that the value of Ftotal
and trapping position depends on the particle size. Therefore, the optical force induced by the
magnetic resonance in MDM fishnet metamaterials could be used for the selection of the size of the
nanoparticles.
Fig. 6. (a) Ftotal exerted on the nanoparticle along the z-axis above ENA for lossy particles "¼"pþiat
Rp¼100 nm at a fixed magnetic resonant wavelength 775 nm; (b) zoom in picture of (a) around
trapping positions.
Fig. 7. (a) Fxon the nanoparticles (Rp¼100 nm and "p¼2:25) for the ENA at different distances
above the structure: Zdist ¼140 nm;160 nm and 180 nm. (b) Fyon the nanoparticles for the ENA.
(c) Fxon the nanoparticles for the RNA.
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In this section, we analyze Ftotal for lossy particles at a fixed wavelength 775 nm in an ENA. We
introduce the loss term into the permittivity "¼"pþi. We choose two different values for the loss
terms ¼0:1 and ¼0:3. In Fig. 6, we present the comparison of Ftotal on nanoparticles
ðRp¼100 nm;"
p¼2:25Þwith and without loss. It is shown that the waveform of Ftotal is similar in
both cases, but the value of Ftotal increases when loss increases. This can be explained by the fact
that the momentum transfer from the photons to the nanoparticles creates additional pushing forces
as the result of nonelastic interactions [10].
We then studied the lateral forces Fxand Fyon the particle in the transverse plane (x–yplane) at
different positions along z-axis. Fig. 7(a)–(b) show the lateral force Fxand Fyfor the ENA, and
Fig. 7(c) shows the lateral force Fxfor the RNA at different distances above the structure:
Zdist ¼140 nm;160 nm;and 180 nm. Note that Fx¼Fyin the RNA due to its symmetrical
geometry. The numerical investigation has revealed that the lateral force distribution takes on a
sinusoidal profile. At any location, the lateral force points toward the center of the fishnet, making
the particle bonded to the beam axis [45]. It also shows that the value of the lateral force increases
as the particle is placed closer to the surface of the fishnet structure. In particular, the lateral force in
the ENA is stronger than for the RNA.
4. Conclusion
In conclusion, we have demonstrated the possibility of realizing pushing and dragging forces using
MDM fishnet metamaterials in the visible regime. It has been shown that an ENA exhibits a greater
optical force than an RNA due to the stronger magnetic dipole resonance. In addition, the total
optical force Ftotal induced by the structures has different signs, which can be used for optical
trapping and manipulation of dielectric nanoparticles with and without loss. In particular, the
strength of the optical force and trapping position depends on the nanoparticle size, therefore
fishnet metamaterials embedded in MDM films can be applied to size selection of nanoparticles.
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