Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy
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Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy
Brynmor J. Davis and P. Scott Carney
University of Illinois at Urbana-Champaign
Optical Characterization and Nanophotonics Laboratory Journal ClubBoston University, December 3 2007
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
• Motivation and background
• The microscope (forward model)
• Data processing (inverse problem)
• Numerical simulations
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Fire Opal
Stained Glass
commons.wikimedia.org/wiki/Image:Koelner_Dom_-_Bayernfenster_04.jpg
www.minerals.net/mineral/silicate/tecto/quartz/images/opal/mexfire3.htm
Size-Dependent PropertiesNanorods - TEM image Extinction Spectra
Oldenburg et al. - Opt. Express, 14 (2006) 6724
Smith et al. - Science, 305 (2004) 788
Metamaterials
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
We aim to determine the nanoparticle polarizability tensor as a function of wavelength.
Patra et al. - App. Phys. Lett., 87 (2005) 101103
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Px
Py
Pz
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥=
α xx α xy α xz
α xy α yy α yz
α xz α yz α zz
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Ex
Ey
E z
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Defined by 6 Parameters
Assumptions
• Particle small compared to
• Particle isolated spatially
• Linear, coherent scattering characterizedFluorescenceRamanSHG, THG
Induced Dipole Moment Polarizability
ElectricField
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
A coherent confocal microscope is sensitive to the linear polarizability, can be spectrally multiplexed and is “standard”.
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Coherent confocal microscopes are highly sensitive and produce data dependent on particle orientation.
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Single fluorescent molecules can be characterized as dipoles and their orientation inferred from far-field intensity measurements.
Measured Theoretical
PSFs vary with dipole orientation
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
We aim to show the feasibility of estimating the particle position and full tensor polarizability as a function of wavelength.
Mock et al. - J. Chem. Phys., 116 (2002) 6755
Measuring the full polarizability removes assumptions regarding particle shape
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
• Motivation and background
• The microscope (forward model)
• Data processing (inverse problem)
• Numerical simulations
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Interference with a reference beam allows the collection of data sensitive to the electric field.
€
I r,ω( ) = μE r( ) ω( ) + E s( ) r,ω( )2
= μE r( ) ω( )2
+ μE r( ) ω( )[ ]H
E s( ) r,ω( ) + E s( ) r,ω( )[ ]H
μE r( ) ω( ) + E s( ) r,ω( )2
Data
ReferenceScattered Field
Constant Background Autocorrelation
Complex DataConjugate Data
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S r,ω( ) = μE r( ) ω( )[ ]H
E s( ) r,ω( )
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
The desired complex data can be isolated with simple processing.
€
I r,ω( ) = μE r( ) ω( ) + E s( ) r,ω( )2
= μE r( ) ω( )2
+ μE r( ) ω( )[ ]H
E s( ) r,ω( ) + E s( ) r,ω( )[ ]H
μE r( ) ω( ) + E s( ) r,ω( )2
SubtractInsignificant
Complex DataRemove via Hilbert
transform
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S r,ω( ) = μE r( ) ω( )[ ]H
E s( ) r,ω( )
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
A beam shaper is used to give a beam with diverse polarization components.
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E b( ) sx,sy ,ω( ) = V sx ,sy,ω( )E r( ) ω( )
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E xb( )
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Eyb( )
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
A high-aperture lens gives many propagation directions and therefore many polarization states in the field.
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E b( ) sx,sy ,ω( ) = V sx ,sy,ω( )E r( ) ω( )
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E l( ) sx,sy ,ω( ) = A sx ,sy,ω( )E r( ) ω( )
Richards and Wolf - Proc. Roy. Soc. London A, 253 (1959) 358
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Exl( )
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E yl( )
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E zl( )
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
The field in the focal region is found by integrating the incident rays in an angular spectrum.
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E b( ) sx,sy ,ω( ) = V sx ,sy,ω( )E r( ) ω( )
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E l( ) sx,sy ,ω( ) = A sx ,sy,ω( )E r( ) ω( )
Richards and Wolf - Proc. Roy. Soc. London A, 253 (1959) 358
€
g ′ r − r,ω( ) = F ′ r − r,ω( )E r( ) ω( )
= − ik2π
E l( ) sx ,sy,ω( )sz sx ,sy( )
∫ e iks⋅ ′ r −r( )dsx dsy
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
The resulting focal fields display significant fields in all directions.
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gx r( )2
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gy r( )2
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gz r( )2
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z = 0
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z = 2λ
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z = λ
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g ′ r − r,ω( ) = F ′ r − r,ω( )E r( ) ω( )
= − ik2π
E l( ) sx ,sy,ω( )sz sx ,sy( )
∫ e iks⋅ ′ r −r( )dsx dsy
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Exl( )
€
Eyl( )
€
E zl( )
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
The scattered field can then be propagated back to the detector.
Scattering produces sources
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k 2α ′ r ,ω( )g ′ r − r,ω( )
Which leads to a scattered field
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E s( ) r,ω( ) = k 2 F T ′ r − r,ω( )∫ α ′ r ,ω( )g ′ r − r,ω( )d3 ′ r
Recall the data expression
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S r,ω( ) = μE r( ) ω( )[ ]H
E s( ) r,ω( )
And assuming a linearly polarized reference
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S r,ω( ) = μk 2 gξ ′ r − r( )gβ ′ r − r( )∫ α ξβ ′ r ,ω( )ξβ∑ d3 ′ r
2D scanning gives z-dependent PSFs:
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S ρ,ω( ) = hξβ ρ;z,ω( )∗∫ α ξβ ρ;z,ω( )ξβ∑ dz
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Diverse PSFs/OTFs mean each component of the polarizability produces a different signature in the data.
OTFs at z=0
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xx
€
xy
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xz
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yz€
yy
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zz
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hξβ ρ;z,ω( ) = μk 2gξ ρ;z,ω( )gβ ρ;z,ω( )
PSF in terms of the focused field
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gx r( )2
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gy r( )2
€
gz r( )2
€
E xl( )
€
E yl( )
€
E zl( )
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
• Motivation and background
• The microscope (forward model)
• Data processing (inverse problem)
• Numerical simulations
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Assuming a single isolated scatterer, the polarizability and position can be estimated by minimizing a cost function.
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α ′ r ,ω( ) = α ω( )δ ′ r − rp( )
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C α ω( ),rp( ) = ˜ S q,ω( ) − ˜ h ξβ q;zp,ω( )e−i qx x p +qy y p( )α ξβ ω( )
ξβ∑
Prior knowledge of the polarizability
Parameter estimation using a cost function
Cost Fourier-Domain Data
OTF at Particle Plane
From Lateral Position
PolarizabilityParameters to Estimate
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Near the focal plane each OTF can be approximately characterized by one magnitude and one phase function.
OTF Magnitudes OTF Phases
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z = 0
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z = .5λ
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z = .5λ
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z = .25λ
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z = λ
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z = λ
€
zz€
xx
€
xy
€
xz
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
The approximation makes it easy to repeatedly calculate the cost.
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C α ω( ),rp( ) = ˜ S q,ω( ) − ˜ h ξβ q;zp,ω( )e−i qx x p +qy y p( )α ξβ ω( )
ξβ∑
€
C α ω( ),rp( ) ≈ ˜ S q,ω( ) − ˜ H ξβ q;ω( )e−i qx x p +qy y p +φξβ q;ω( )kz p( )α ξβ ω( )ξβ∑
Magnitude Function Phase Function
Minimization is linear (easy) in polarizability and nonlinear in position
Cost Fourier-Domain Data
OTF at Particle Plane
From Lateral PositionPolarizability
Parameters to Estimate
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
The Nelder-Mead algorithm is used to iteratively minimize over the three position variables.
en.wikipedia.org/wiki/Image:Nelder_Mead2.gif
Nelder and Mead - The Computer Journal, 7 (1965) 308
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
• Motivation and background
• The microscope (forward model)
• Data processing (inverse problem)
• Numerical simulations
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Simulated data can be created from a given polarizability and particle position.
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α =.433+ .633i .137 − .380i −.308 + .424i.137 − .380i −.540 + .164i −.096 − .293i
−.308 − .424i −.096 − .293i −.087 + .185i
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
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r = λ−1.67−1.24−.088
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
No Noise SNR=13dB SNR=4dB
Real Part
Imaginary Part
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
The reconstruction algorithm matches data in the Fourier domain.
€
r = λ−1.67−1.24−.088
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥€
α =.433 + .633i .137 − .380i −.308 + .424i.137 − .380i −.540 + .164i −.096 − .293i
−.308 − .424i −.096 − .293i −.087 + .185i
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
ˆ α =.417 + .622i .137 − .401i −.339 + .405i.137 − .401i −.543 + .170i −.031− .279i
−.339 − .405i −.031− .279i −.111 + .168i
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
r = λ−1.66−1.25−.085
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
No Noise
SNR=13dB
Given Parameters
Estimated Parameters
ReconstructionFrom Noisy Data
Magnitude Phase
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Monte Carlo simulations show performance degrades with noise and distance from the focal plane.
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Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007
Summary
• The nanoparticle’s position and wavelength-dependent linear polarizability can be accurately estimated.
• Estimates are from a single coherent confocal spectral image.
• The prior assumption of one small isolated scatterer is required.
• The method relies on polarization diversity in the focused field.
• The method is robust to noise and defocus.
Contact me: [email protected]