1. 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 3 2007

2. 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

3. 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 Properties Nanorods - TEM image Extinction Spectra Oldenburg et al. - Opt. Express, 14 (2006) 6724 Smith et al. - Science, 305 (2004) 788 Metamaterials

4. 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 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

5. 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”.

6. 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.

7. 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

8. 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

9. 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

10. 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. Data Reference Scattered Field Constant Background Autocorrelation Complex Data Conjugate Data

11. Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007 The desired complex data can be isolated with simple processing. Subtract Insignificant Complex Data Remove via Hilbert transform

12. 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.

13. 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. Richards and Wolf - Proc. Roy. Soc. London A, 253 (1959) 358

14. 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. Richards and Wolf - Proc. Roy. Soc. London A, 253 (1959) 358

15. Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007 The resulting focal fields display significant fields in all directions.

16. 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 Which leads to a scattered field Recall the data expression And assuming a linearly polarized reference 2D scanning gives z-dependent PSFs:

17. 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 PSF in terms of the focused field

18. 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

19. 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. 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

20. 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

21. Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007 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

22. 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

23. 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

24. 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. No Noise SNR=13dB SNR=4dB Real Part Imaginary Part

25. Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007 The reconstruction algorithm matches data in the Fourier domain. No Noise SNR=13dB Given Parameters Estimated Parameters Reconstruction From Noisy Data Magnitude Phase

26. 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.

27. 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: bryn@uiuc.edu