1. Congresso del Dipartimento di Fisica Highlights in Physics 2005 11–14 October 2005, Dipartimento di Fisica, Università di Milano An application of the Optical Theorem to the sizing of sub-wavelength particles M.A.C. Potenza *, M. Giglio * * Dipartimento di Fisica, Università di Milano From the general theory of scattering, the phase of the scattered wave depends on the particle dimension The removed power is negligible. Nevertheless the interference pattern is visible, and the spherical wave is measured !! Incident plane wave Scattered wave Interference pattern (paraxial approximation) Small particle (with respect to the wavelength) The forward s cattered and transmitted waves are in phase Large particle (the size of the wavelength or larger) The forward scattered and transmitted waves are in quadrature The removed power is appreciable by the intensity at the centre of the pattern. Light scattering from a single particle transmitted power = integral of the interference pattern ABSTRACT We present a novel technique based on the Optical Theorem (OT) for the determination of the size of scattering particles in the sub-wavelength range. When a plane wave impinges on a particle, a scattered spherical wave is generated. The scattered amplitude is in general made up by both in phase and in quadrature components. The OT states that the scattering cross section that describes the integral of the scattered intensity at any angle is proportional to the amplitude of the quadrature component at zero scattering angle. This remarkable result is the consequence of conservation arguments that require that the interference between the scattered wave and the incoming wave should exactly account for the power loss of the incoming beam due to scattering of power in different directions. We show an application of the method in the realistic case where an assembly of particles is present under the incoming beam. The scattered radiation in the far field is then a speckle field, with the intensity that fluctuates as a consequence of the stochastic interference between individual scattered waves. If the scattered intensity is collected fairly close to the scattering sample, and the incoming beam is wide enough, then low visibility speckle field is generated due to the interference between the weak scattered fields and the powerful transmitted beam. The method relies on the statistical analysis of these low visibility fringes due to the heterodyning between scattered field and the transmitted beam that acts as a self referencing local oscillator. The statistical processing of the instantaneous speckle distribution consists of the evaluation of the two dimensional power spectrum of the intensity distribution. It will be shown that the spectrum exhibits a multiple zero structure, from which an accurate estimate of the amplitude of the quadrature term can be performed. Spherical, calibrated polystyrene colloidal particles have been used. The results have been compared with the theoretical predictions based on Mie scattering functions and the OT. The data show a continuous variation of the phase of the scattered amplitude from zero to p/2 as the particle diameter is changed from well subwavelength values to many microns sizes, as expected from the OT as one moves from the Rayleigh scattering regime to the diffraction regime. Excellent accuracy for the subwavelength diameters determination is reported. Extinction factor Q ext = σ/πa 2 Complex scattered field (normalized by πa 2 ) Phase lag of the forward scattered wavefront Relative refractive index: m = 1.55 (typical for airborne particles) Relative refractive index: m = 1.18 (polystyrene in a water suspension) diameter 0.1-4 um x = ka diameter 0.1-4 um x = ka diameter 0.1-2 um diameter 0.1-4 um diameter 0.1-2 um Re[S(0)] Im[S(0)] diameter 0.1-4 um Re[S(0)] Im[S(0)] EXPERIMENTAL APPARATUS AND DATA ANALYSIS The measurements have been performed by the self referencing optical system scketched in the figure. A plane wave transilluminates a monodisperse, dilute particle suspension and the transmitted beam falls over the plane of a CCD camera, acting as a local oscillator for the scattered wavefornts. The intensity distribution is the stochastic sum of the intensities of all the elementary interference patterns generated by each particle in the sample. 100 nm 300 nm 1000 nm 300 nm “An experiment by which this spherical wave can be observed is impossible, for a telescope (…) sees the primary and secondary source in the same direction: their images coincide” (Van de Hulst) Light scattering from a collection of particles The intensity profile is the sum of the elementary patterns 100 nm600 nm2000 nm Single interference pattern Power spectrum of the speckle field Radial profile of the power spectum Despite the speckled appearance, the power spectrum of the intensity distribution provides the information about the phase Laser source Collimating lens Sample cell CCD sensor EXPERIMENTAL RESULTS Comparison of the positions of the extrema in the radial profiles of the measured power spectra to the expected values. Water suspensions of 400 nm, 600 nm, 1000 nm, 2000 nm in diameter have been used in a cell 2 mm thick. Different sample-sensor distances have been used. Light is transmitted E0E0 If E 0 >> E S :I = | E 0 | 2 + E 0 E S * + E 0 * E S + E S 2 scattered ESES by the sample The signal E S is heterodyned by the transmitted field E 0 E S being stochastic, by averaging over a number of realizations of the speckle field I i (x,y), we have: = | E 0 | 2 = I 0 that is the transmitted field plus the static stray light ! Good accordance to the expected positions of the maxima and minima. No free parameters are present to fit the dephasing values. The dephasing of the scattered waves depends on the particle size accordingly to the Mie Theory. The method has been filed for PCT patent in 2005 ( University of Milan ) Diameter 2000 nm z = 29 mm Diameter 1000 nm z = 29 mm Diameter 600 nm z = 29 mm Diameter 400 nm z = 147 mm Diameter 2000 nm z = 62 mm Diameter 600 nm z = 62 mm A few particles Many particles (speckles) The extrema positions depend on the dephasing value. Due to the interference, the dephasing of the scattered wave reduces the field amplitude in the forward direction. This accounts for the power scattered away: conservation of the energy ! The particle cross section is: OPTICAL THEOREM