Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy

1. Brynmor J. Davis and P. Scott Carney University of Illinois at Urbana-Champaign Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy Optical Characterization and Nanophotonics Laboratory Journal Club Boston University, December 3 2007

2. 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. 3 2007

3. Size-Dependent Properties Nanorods - TEM image Extinction Spectra Fire Opal www.minerals.net/mineral/silicate/ tecto/quartz/images/opal/mexfire3.htm Stained Glass Oldenburg et al. - Opt. Express, 14 (2006) 6724 Metamaterials Smith et al. - Science, 305 (2004) 788 commons.wikimedia.org/wiki/Image:Koelner_Dom_-_Bayernfenster_04.jpg Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

4. We aim to determine the nanoparticle polarizability tensor as a function of wavelength. Defined by 6 Parameters • Assumptions • Particle small compared to  • Particle isolated spatially • Linear, coherent scattering characterized • Fluorescence • Raman • SHG, THG Induced Dipole Moment Electric Field Polarizability Patra et al. - App. Phys. Lett., 87 (2005) 101103 Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

5. 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. 3 2007

6. 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. 3 2007

7. Single fluorescent molecules can be characterized as dipoles and their orientation inferred from far-field intensity measurements. PSFs vary with dipole orientation Measured Theoretical Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

8. We aim to show the feasibility of estimating the particle position and full tensor polarizability as a function of wavelength. Measuring the full polarizability removes assumptions regarding particle shape Mock et al. - J. Chem. Phys., 116 (2002) 6755 Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

9. 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. 3 2007

10. Interference with a reference beam allows the collection of data sensitive to the electric field. Reference Scattered Field Data Constant Background Autocorrelation Conjugate Data Complex Data Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

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

12. 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. 3 2007

13. 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. 3 2007

14. 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. 3 2007

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

16. 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. 3 2007

17. Diverse PSFs/OTFs mean each component of the polarizability produces a different signature in the data. PSF in terms of the focused field OTFs at z=0 Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

18. 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. 3 2007

19. 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 From Lateral Position Cost Fourier-Domain Data Polarizability OTF at Particle Plane Parameters to Estimate Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

20. 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. 3 2007

21. The approximation makes it easy to repeatedly calculate the cost. Cost From Lateral Position Fourier-Domain Data Polarizability OTF at Particle Plane Parameters to Estimate Phase Function Magnitude Function Minimization is linear (easy) in polarizability and nonlinear in position Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

22. 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. 3 2007

23. 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. 3 2007

24. Simulated data can be created from a given polarizability and particle position. Real Part Imaginary Part No Noise SNR=13dB SNR=4dB Davis & Carney, Nanoparticle Polarizability Determination Using Coherent Confocal Microscopy, Boston University, Dec. 3 2007

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

26. 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. 3 2007

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