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Timestamp: 2019-04-25 16:56:18+00:00

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Micrometer sized particles can be accurately characterized using holographic video microscopy and Lorenz-Mie fitting. In this work, we explore some of the limitations in holographic microscopy and introduce methods for increasing the accuracy of this technique with the use of multiple wavelengths of laser illumination. Large high index particle holograms have near degenerate solutions that can confuse standard fitting algorithms. Using a model based on diffraction from a phase disk, we explain the source of these degeneracies. We introduce multiple color holography as an effective approach to distinguish between degenerate solutions and provide improved accuracy for the holographic analysis of sub-visible colloidal particles.
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Fig. 1 Schematic for models of holographic microscopy. (a) Thin disk model for hologram formation where incident light, represented by green arrows, impinges on the particle represented by the orange disk. The transmitted light is delayed by the disk, and then diffracts and interferes with the incident light. This interference creates the hologram represented by the ringed pattern below the plane of the disk. The hologram shown is data collected from a sample of 1.5 µm polystyrene particles suspended in DI water and illuminated by a 532nm laser. (b) The particle as a Fabry-Perot etalon where the orange cylinder represents a particle and the green arrows represent light that reflects from or transmits through the particle.
Fig. 2 Experimental evidence of degenerate solutions. Images (b–d), marked “Data”, are experimentally measured holograms of 1.5 µm diameter polystyrene spheres. Each hologram was measured with a different illumination wavelength, 450 nm, 532 nm and 635 nm respectively. Below, images (e–g), marked “Sphere”, represent the best fit using Lorenz-Mie theory to the corresponding experimental hologram, with an initial guess in diameter, dp, and refractive index, np, at the expected values of 1.5 µm in diameter and 1.6 in refractive index. In (a), additional local minima are found at slightly different size and significantly different refractive index values. The color of the points corresponds to the illumination wavelength, blue: 450 nm, green: 532 nm, and red: 635 nm. For each point in (a), we plot the radial profile, or azimuthal average, of the theoretical hologram corresponding to a sphere with such size and refractive index values in (h–j). The theoretical radial profiles are grouped and colored according to their wavelength. Each theoretical radial profile is compared to the corresponding experimental radial profile (black curve) for that wavelength.
Fig. 3 Comparison of fits using Fabry-Perot disk model to experimental holograms. Images (b–d), marked “Data”, are experimentally measured holograms of 1.5 µm diameter polystyrene spheres repeated from Fig. 2. Below, images (e–g), marked “Disk”, represent the best fit using the transparent disk model to the corresponding experimental hologram, with an initial guess in size and refractive index near the expected values of 1.5 µm in size and 1.6 in refractive index. Similar to Lorenz-Mie theory, additional local minima are found at slightly different size and significantly different refractive index values. These points are shown in (a) as stars with the corresponding points from Lorenz-Mie theory replotted from Fig. 2(a). The color of the points corresponds to the illumination wavelength, blue: 450 nm, green: 532 nm, and red: 635 nm. For each point in (a), we plot the radial profile, or azimuthal average, of the theoretical hologram corresponding to a disk with such size and refractive index in (h–j). The theoretical radial profiles are grouped and colored according to their wavelength. Each theoretical radial profile is compared to the experimental radial profile, colored black, for that wavelength.
Fig. 4 Comparison of errors of Lorenz-Mie theory (solid lines) and the disk model (dotted lines) as a function of refractive index for two PS spheres: 1.5 µm (left column) and 9.7 µm (right column). In each panel, the normalized sum of the squared differences between the theoretical hologram and the experimental hologram are plotted versus the refractive index while all other fit parameters are held constant. Each row corresponds to data at a specific wavelength: (a–b) 450 nm, (c–d) 532 nm, and (e–f) 635 nm.
Fig. 5 Experimental distributions of particle hologram fits to Lorenz-Mie theory as a function of refractive index. Panels (a) and (b) show the fit distributions of 9.7 µm PS particles measured in 450 nm and 532 nm illumination respectively. The vertical dashed lines represent expected position of degenerate peaks. After the true peak is determined, the particles are refit using the true values as starting parameters, and the resulting narrow distributions are plotted in (c) for 450 nm and (d) for 532 nm. In panels (c) and (d), the global minimum is indicated by the vertical gray line.
(3) b ( r ) | E s ( r ) | 2 E 0 2 = | x ^ + exp ( i k z p ) f s ( k ( r − r p ) ) | 2 .
(4) Δ ϕ = 2 π λ ( n p − n m ) d c .
(8) n N = N λ d p + n p .
(10) t c = E t E i = ( 1 − R ) exp ( i Δ ϕ ) 1 − R exp ( i 2 Δ ϕ ) .
(11) E ( r ) = E 0 ( r ) + E d ( r ) ( t c − 1 ) .
(12) Norm Sq . Diff . = ∑ i ( I ( r i ) − I model ( r i ) ) 2 ∑ i | I ( r i ) − 1 | .

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