Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy

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

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


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

http://www.minerals.net/mineral/silicate/


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

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α xx α xy α xz

α xy α yy α yz

α xz α yz α zz

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Ex

Ey

E z

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Defined by 6 Parameters

Assumptions

• Particle small compared to

• Particle isolated spatially

• Linear, coherent scattering characterizedFluorescenceRamanSHG, THG

Induced Dipole Moment Polarizability

ElectricField


A coherent confocal microscope is sensitive to the linear polarizability, can be spectrally multiplexed and is “standard”.


Coherent confocal microscopes are highly sensitive and produce data dependent on particle orientation.


Single fluorescent molecules can be characterized as dipoles and their orientation inferred from far-field intensity measurements.

Measured Theoretical

PSFs vary with dipole orientation


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







Interference with a reference beam allows the collection of data sensitive to the electric field.

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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( )


The desired complex data can be isolated with simple processing.

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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( )


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( )


A high-aperture lens gives many propagation directions and therefore many polarization states in the field.

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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( )


The field in the focal region is found by integrating the incident rays in an angular spectrum.

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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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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


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( )

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Eyl( )

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E zl( )


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


Diverse PSFs/OTFs mean each component of the polarizability produces a different signature in the data.

OTFs at z=0

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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

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gz r( )2

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E xl( )

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E yl( )

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E zl( )







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


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

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xx

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xy

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xz


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∑

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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


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







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

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No Noise SNR=13dB SNR=4dB

Real Part

Imaginary Part


The reconstruction algorithm matches data in the Fourier domain.

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r = λ−1.67−1.24−.088

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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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ˆ α =.417 + .622i .137 − .401i −.339 + .405i.137 − .401i −.543 + .170i −.031− .279i

−.339 − .405i −.031− .279i −.111 + .168i

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r = λ−1.66−1.25−.085

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No Noise

SNR=13dB

Given Parameters

Estimated Parameters

ReconstructionFrom Noisy Data

Magnitude Phase


Monte Carlo simulations show performance degrades with noise and distance from the focal plane.


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]

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Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy