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 Club Boston University, December

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Motivation and background The microscope (forward model) Data processing (inverse problem) Numerical simulations

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Fire Opal Stained Glass commons.wikimedia.org/wiki/Image:Koelner_Dom_-_Bayernfenster_04.jpg tecto/quartz/images/opal/mexfire3.htm Size-Dependent Properties Nanorods - TEM image Extinction Spectra Oldenburg et al. - Opt. Express, 14 (2006) 6724 Smith et al. - Science, 305 (2004) 788 Metamaterials

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec We aim to determine the nanoparticle polarizability tensor as a function of wavelength. Patra et al. - App. Phys. Lett., 87 (2005) Defined by 6 Parameters Assumptions Particle small compared to Particle isolated spatially Linear, coherent scattering characterized Fluorescence Raman SHG, THG Induced Dipole Moment Polarizability Electric Field

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec A coherent confocal microscope is sensitive to the linear polarizability, can be spectrally multiplexed and is “standard”.

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Coherent confocal microscopes are highly sensitive and produce data dependent on particle orientation.

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Single fluorescent molecules can be characterized as dipoles and their orientation inferred from far-field intensity measurements. Measured Theoretical PSFs vary with dipole orientation

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec 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

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Motivation and background The microscope (forward model) Data processing (inverse problem) Numerical simulations

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Interference with a reference beam allows the collection of data sensitive to the electric field. Data Reference Scattered Field Constant Background Autocorrelation Complex Data Conjugate Data

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec The desired complex data can be isolated with simple processing. Subtract Insignificant Complex Data Remove via Hilbert transform

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec A beam shaper is used to give a beam with diverse polarization components.

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec A high-aperture lens gives many propagation directions and therefore many polarization states in the field. Richards and Wolf - Proc. Roy. Soc. London A, 253 (1959) 358

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec The field in the focal region is found by integrating the incident rays in an angular spectrum. Richards and Wolf - Proc. Roy. Soc. London A, 253 (1959) 358

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec The resulting focal fields display significant fields in all directions.

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec The scattered field can then be propagated back to the detector. Scattering produces sources Which leads to a scattered field Recall the data expression And assuming a linearly polarized reference 2D scanning gives z-dependent PSFs:

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Diverse PSFs/OTFs mean each component of the polarizability produces a different signature in the data. OTFs at z=0 PSF in terms of the focused field

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Motivation and background The microscope (forward model) Data processing (inverse problem) Numerical simulations

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Assuming a single isolated scatterer, the polarizability and position can be estimated by minimizing a cost function. Prior knowledge of the polarizability Parameter estimation using a cost function Cost Fourier-Domain Data OTF at Particle Plane From Lateral Position Polarizability Parameters to Estimate

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Near the focal plane each OTF can be approximately characterized by one magnitude and one phase function. OTF Magnitudes OTF Phases

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec The approximation makes it easy to repeatedly calculate the cost. Magnitude Function Phase Function Minimization is linear (easy) in polarizability and nonlinear in position Cost Fourier-Domain Data OTF at Particle Plane From Lateral Position Polarizability Parameters to Estimate

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec 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

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Motivation and background The microscope (forward model) Data processing (inverse problem) Numerical simulations

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Simulated data can be created from a given polarizability and particle position. No Noise SNR=13dB SNR=4dB Real Part Imaginary Part

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec The reconstruction algorithm matches data in the Fourier domain. No Noise SNR=13dB Given Parameters Estimated Parameters Reconstruction From Noisy Data Magnitude Phase

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec Monte Carlo simulations show performance degrades with noise and distance from the focal plane.

Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec 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: