Computational study of light stabilized structures comprising of nanoparticles

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Computational study of light stabilized
structures comprising of nano-particles

• C.T. Chan
• Hong Kong University of Science and Technology

Collaborators Light induced forces Jack Ng,
Z.F. Lin, P. Sheng
International Workshop on Computational Methods
for Nanoscale Systems Dec. 11 13, 2006
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Outline

• Force of light on matter
• Use light to organize matter

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Light ? Structure

• Use structured material to manipulate light.
• Can we use light to manipulate structure ?
• Light can induce force to manipulate
  nano-particles into designed structures
• Light has energy, momentum and angular momentum
• Force is small, but significant for very small
  particles
• May be the method of choice to organize nano
  particles

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• Well known
• Photon pressure
• Gradient force (force due to inhomogeneous light
  intensity

Less well known Homogenous light field can
induce long range inter-particle forces to bind
clusters into a stable structure (from multiple
scattering of light) Short range forces due to
near field coupling of resonances
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The optical forcein classical electrodynamics

• Matter modifies the flow of light Maxwell
  Equations

• Light exerts forces on matterLorentz Force
  Equation, the definition of the EM fields

Point Charge
Arbitrary Particle
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The Maxwell Stress Tensor for computing the
optical force
The optical force acting on a particle can be
found by integrating the Lorentz force over the
particles volume
It can be shown, after a lot of mathematics
The motion of the particles are small, therefore
we take time averaged
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Understanding the nature of optical force (using
a dipole as an example)
A neutral particle with size ? illuminated by
an incident field
d ?
will develop an electric dipole moment
Electric dipole polarizability (an intrinsic
property of the particle)
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Understanding the nature of optical force
The optical force acting on a dipole can be
derived most easily by treating the dipole as a
pair of equal and opposite charges
Lorentz Force acting on a point charge
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Electrical force acting on a dipole
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q
Magnetic force acting on a dipole
-q
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Total force acting on a dipole
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The Gradient Force the Scattering and
Absorption Force
The force can be separated into two physically
and mathematically distinct components
(1) The Conservative Gradient Force
Induced by the gradient of amplitude
The electromagnetic free energy
(2) The Nonconservative Scattering and Absorption
Force
Induced by the gradient of phase
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Optical Tweezers
D.G. Grier, Nature 424, 810 (2003).

• Near the focus, gradient force dominates ,
  dielectric particles can be trapped.

• In typical experiments, the optical force
  pico-Newton
• Greater than or at least comparable to weight of
  small particles

• In principle, anything that scatter light can be
  trapped, which includes (1) Metal(2) Atoms(3)
  Bacteria(4) Red blood cells(5) Many more

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Optical Binding

• The first dipole scatters the light
• Modifying the fields striking the second dipoles
• The force acting on the dipoles depend on their
  relative positions!

Optical binding force in the far field
The scattered field from one dipole interact with
the dipole moment on the other particle
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Optical Binding
Optical binding force (in the limit of R-gt8)
Oscillatory force
Long range force(inverse distance),since the
radiationfield decays as inverse distance
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Gradient Forces microspheres move to high
intensity of light
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• Interference of coherent beams makes optical
  standing waves
• Gradient force can organize particles into
  pattern

Burns et al, Science 249, 749 (1990).
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Optical Binding Force

• Homogeneous intensity force due to scattering
• Demonstrated experimentally in 1989. (Burns et
  al, PRL Vol. 63 P.1233 1989)
  
  
  
  

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Attractive/Repulsive optical forces between 2
plastic spheres
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Optical Binding Force
Lorentz Force
Approximate Interaction Energy (far field,
lowest order in 1/r)
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Calculation of optical forces on
nano/micro-spheres cluster MS-MST formalism
Compute EM-field by Multiple scattering theory
Compute time-averaged EM-force Torque by
Maxwell Stress Tensor
Integration can be performed in closed form (in
terms of the scattering coefficients)
Force
Torque
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Formalism (MS-MST)Maxwell Tensor and
Multiple-Scattering Approach

• Advantages
• Efficient (1,000 Mie scatters in a PC), rigorous
  and exact in principle.
• Applicable to any cluster of spheres.
• Not limited to small particle size (comparable to
  wavelength).
• Arbitrary dielectric function (dispersive,
  absorption)
• Coated sphere is ok

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Optical Binding
?2.53 rs0.414 ?m
Radial optical force
Ng et. al., PRB 72, 085130 (2005)
Transverse optical force
Fournier et. al., PRL 63, 1233 (1989)
?
van der Waals force(non-retarded)
Spheres in contact
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• Multiple Stable points exist on both axis
• The force has a angular dependence

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Vibration Frequency Each stable equilibrium is
stable against perturbation,with well defined
vibration frequencies
-Radial force is stronger, higher frequency
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Photonic Molecules ofNano-spheres (krs0.4)
Light-induced organization of nano-particles Refr
active index 1.59 mass density 1050
kg/m3 Wavelength 0.52 micron 2 Beams coming
from z and z direction, x-polarized and in
phase at xy-plane.
Polarization Direction
Note multiple equilibriums for same number of
spheres
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Molecule of nano-spheres bounded by optical
forces

• Ordinary molecules
• Made with atoms
• Scale nanometer
• Bounded by electrostatic forces QM
• States Schrodinger equation
• Forces H-F Theorem
• Small number of stable metastable states

• Optical clusters
• Made with nano-particles
• Scale micro/nano
• Bounded by light exchange of real photons
• States Maxwell equation
• Forces Maxwell-tensor
• Large number of stable metastable states
  quasi-meta-stable states

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Photonic clusters microspheres bound by light
(krs5.0, radius0.414 ?m, ?0.52 ?m)

• Incident field is homogeneous
• Stability solely induced by optical binding

Ng et. al., PRB 72, 085130 (2005)
Polarization Direction
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Multiple scattering induced optical force can
bind microsphere into clusters with distinct
geometries and exotic dynamical characteristics.

• Long range force
• Multiple equilibriums
• Pinned in orientation
• Drifting equilibrium (if no inversion symmetry)
• Non-symmetric force matrix Complex dynamics

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Stability Analysis

• The eigenvalues of the force constant matrix
  gives the stability of a photonic molecule
• We have neutral, stable, unstable, and drifting
  and quasi-stable
• Drifting All molecules have same force, move as
  an entity
• Quasi-stable Force not conservative (open
  system), force matrix can have pairs of complex
  eigenvalues, conditional stable in presence of
  friction
• Full molecular dynamics needed in some cases

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Dynamics of photonic molecule with complex
frequency
Force matrix can have complex eigenvalues Compli
cated dynamical behavior depending on damping of
the environment
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Polystyrene Spheres R 0.331 micron
kR 4.0 Intensity 1010 W/m2
Wavelength0.52 micron B-critical43.1
pN/ms-1 Time interval 1 ms
Small damping Quasi-stable
No damping Unstable
Large damping Stable
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Resonant optical force at the whispering gallery
mode of microsphere-cluster
Microsphere High Q-factor EM Cavity

• High-Q microsphere-cluster

Incident light at resonant frequency
--------------------------------------------------
---------------------------- Very strong
interparticle optical force
Opt. Lett. 30, 1956 (2005)
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Geometric interpretation of the WG mode Light
trapped inside a sphere by total internal
reflection. Very strong field near the
surface.
Sharp resonance peaks Whispering Gallery (WG)
modes. At WG mode Surface field can be enhanced
by gt1000 times. But photon pressure is enhanced
by 25 ONLY. Reason EM fields are equally
strong on both sides of a sphere, their effects
cancel.
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Resonant radiation pressure in dielectric
microspheres

• Resonant radiation pressure has been observed in
  optical levitation.
• Despite the high-Q, the optical force is
  enhanced only by 30-50.
• We shall see that an inhomogeneous plane wave
  (or an evanescent wave) can induce an
  extremely strong force.

A. Ashkin, Science 210, 1081 (1980).
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Strong resonant optical force excited by
evanescent waves
Force along the incident wavespropagating
direction
e2.53 a 2.5 µm
Evanescent waves (Inhomogeneous plane wave)
Light Intensity (at the bottom of the
sphere) 104 W/cm2
Plane waves
Light Intensity 104 W/cm2
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Evanescent wave excitation for size selective
manipulation of microspheres

• Sharp WGM ? accurate size selective
  manipulation is possible (better than 0.05 for a
  5-µm-diam sphere)

At low intensity, only those with a particular
size, which happens to be at resonance, will
experience a noticeable optical force.

• Pick up microspheres that has exactly the size
  you want.
• Sorting microsphere by size, in an automatic,
  parallel, accurate, and high throughput
  manner.

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Resonant optical force
Intensity
Bonding mode Strong attraction Q-factor 104
Anti-bonding mode Strong repulsion Q-factor 103
5-micron-diameter Weight 1 pN Plane
wave, Light Intensity 104 W/cm2
2 spheres ?2.5310-4i Realistic e.g. glass
Top sphere
Bottom sphere
Hybridization
Single Sphere ?2.53
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Robust against size dispersion
2 Identical Spheres 2 X (5-?m-diameter)
2 Spheres differ by 2 in diameter
?2.5310-4i Light Intensity (Plane wave) 104
W/cm2
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The effect of absorption and gain
Small gain, Q7 x 105. (using an simple
anti-absorption model)
No loss, Q5.6 x 105.
Small loss, Q1.5 x 104
Re(e)2.53 a 2.5 µm
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Recent Experiments observing resonances

• Microsphere cluster
• Very high quality factor, high internal field

Probe

• Resonant fluorescence in bisphere.
• T. Mukaiyama et. al., Phys. Rev. Lett. 82, 4623
  (1999).Y.P. Rakovich et. al., Phys. Rev. A 70,
  051801 (2004).

Polystyrene sphere

• Lasing in the resonant modes of a bisphere.
• Y. Hara et al, Opt. Lett. 28, 2437 (2003).

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Extended structures

• Light can bind particles into a cluster, can it
  stabilize an extended structure ?
• In 1D Yes, if the particle is small enough
• These structures have special vibrational
  excitations

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Bain et. al.Polystyrene particles formed a
linear chain upon illumination by
counterpropagating waves. The spacing of the
chain matches that of the laser interference
patterns.
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C.D. Mellor et. al., Opt. Exp. 14, 10079
(2006). Colloidal particles formed arrays as a
result of laser induced optical force. The
geometry of the array depends on particle size
and light polarization.
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Garces-Chavez et. al., PRB 73, 085417 (2006). By
increasing the incident light power, the optical
forces dominate and bind the particles into an
array.
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Unstable regime
For point particles, an infinite chain is
stable For finite size particles, there is a
maximum number of spheres that can be made stable
Stable regime
Sphere size normalized to wavelength ?
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Unstable regime
Stable regime
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Localized vibrational modes in optically-bound
structure of extended size

• Light can stablize an extended 1D
  optically-bound lattice
• Lattice constant defined by external field
• Optical binding provides transverse stability
• Such lattice support intrinsically localized
  modes (no defect, no disorder, no non-linearity)

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Characteristic vibration modes
I0106 W/cm2
?2.53 radius ? /10
Characteristic vibrational modes derived from
linear stability analysis
?/2260 nm
Diagonalization
Figures not draw to scale !!
Characteristic vibration amplitude and frequency
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Inverse Participation Ratio an indication of
the no. of participating particles

• Optically-bound structures are more localized
• P.E. model captures the salient features of the
  MS-MST calculation.

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Localized modes in 2D structures

• Localization induced by the long ranged optical
  binding
• Not restricted to the particular geometry or
  incident wave

Vibration Amplitude of individual particle
Localized on the boundary
Localized in center
?2.53 rs0.414 ?m ?0.52 ?m
Polarization Direction
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Summary

• Formalism (similar to ab-initio dynamics) has
  been developed for light-induced dynamics
• Light can be used to organize nano-particles into
  stable configuration (optical matter)
• The dynamics and lattice dynamics of optical
  matter are interesting and different from
  electronic material
• Local modes for periodic system

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Thank You
Acknowledgement Funding HK-RGC Collaborators L
ight induced forces Jack Ng, Z.F. Lin, P. Sheng
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Formalism (MT-MS)Maxwell Tensor and
Multiple-Scattering Approach
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Formalism (MT-MS)Maxwell Tensor and
Multiple-Scattering Approach
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Only eigenmodes with azimuthal quantum number
equal to 1 or -1 are excited.
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Quasi-stable photonic cluster

• Light-matter interaction nonconservative
• quasi-stable photonic clusters gt vibrational
  frequency ??Complex Re( ?2)gt0stability depends
  damping

Conclusion is qualitatively insensitive to the
form of damping

• Dynamics simulations

• bltbcritical unstable
• Particles go into another more stable structure

• bgtbcritical stable
• Particles return to equilibrium

• b bcritical
• Particles will be orbiting around the
  equilibrium.
• The energy loss by damping is counter-balanced
  by light absorption.