Nonparaxial Mie Theory of Image Formation in Optical Microscopes and Characterization of Colloidal Particles

We derive an explicit partial-wave (Mie) series for the image of a dielectric microsphere collected by a typical infinity-corrected microscope. We model the propagation of the illumination and scattered vector fields through the optical components of the microscope by using the angular-spectrum theorem with the help of Wigner rotation matrix elements, allowing us to identify the contribution from spin-orbit helicity reversal. We consider a high numerical aperture objective well beyond the validity range of the paraxial approximation. The spherical aberration introduced by refraction at the planar interface between the sample and the glass slide is fully taken into account. By comparing our theoretical model with images of colloidal particles placed at different positions with respect to the objective focal plane, we characterize their radii and refractive index. We employ polystyrene microspheres with a known refractive index in order to fit the transverse attenuation length describing the transmission loss of the scattered field. As an application, we measure the radius and refractive index of individual silica beads. We compare the result for the radius with an independent measurement using high-resolution scanning electron microscopy. To validate the result for the refractive index, we develop a second method, independent of the theoretical model, based on the image contrast in glycerin-water solutions. In all cases we find very good agreement between our method and the validation procedures. In addition, the nonparaxial theory provides a reliable description of the images found for all focal-plane positions and for both polystyrene and silica microspheres. Our approach allows a common optical microscope to be used to measure the refractive index and radius of spherical particles covering the entire size range from the Rayleigh regime to the ray optics one.

Figure 1

Propagation of the field scattered by a dielectric microsphere (radius $a$) through the microscope optical train. The scattering angle $\alpha$ in the host aqueous medium is related to the angle $\theta$ in the glass medium by Snell’s law of refraction. OL (focal length $f$) and TL (focal length $f^{\prime }$) are the objective and tube lenses. The distance $D$ between them satisfy $D=f+f^{\prime }$. $S_1-S_4$ are spherical reference surfaces as defined by the sine condition. We calculate the total electric energy density at a point ${\mathbf{r}}_t$ in the image space of TL.

Figure 2

From left to right, the image of the polystyrene particle (attached to the glass slide) is brought out of focus by driving the microscope stage downwards. As a consequence, the objective focal plane (red) is displaced upwards with respect to the glass slide, as illustrated by (a)–(c). The corresponding experimental and simulated images are shown in (d)–(f) and (g)–(i), respectively. Plots of the theoretical (line) and experimental (circles) image contrast versus the distance to center $\rho$ are shown in (j)–(l). The scale is the same for all images and is indicated by a horizontal bar ($5\mu \mathrm{m}$) in (e).

Figure 3

Images and contrast plots of a silica microsphere attached to the glass slide. Same conventions as in Fig. 2. The scale bar in (e) corresponds to $5\mu \mathrm{m}$ also as in Fig. 2. In contrast with Fig. 2, the initial reference image (d) corresponds to a focal-plane position below the glass slide as indicated in (a).

Figure 4

Glycerin method as an alternative measurement of the refractive index of silica microspheres. (a) Experimental values of $\mathcal{M}$ (circles with error bars) versus glycerin concentration. A smooth fit to the data (solid line) is also shown. Inset: enlargement showing the experimental data and a quadratic fit weighting the error bars for the determination of the concentration $[\mathrm{gly}{]}_{\min }$, and respective uncertainty. (b) Images of the silica microsphere in pure water and in glycerin-water solutions of $20\mathrm{\%}$, $60\mathrm{\%}$, and $90\mathrm{\%}$ concentrations. The images correspond to focal-plane positions such as to minimize the integrated squared contrast as required by the definition of $\mathcal{M}$ [see Eq. (17)]. The scale is the same in all images and is indicated by the scale bar of $1\mu \mathrm{m}$. The refractive index is obtained after substitution of $[\mathrm{gly}{]}_{\min }$ into Eq. (16).

Figure 5

(a) HSEM image of silica microspheres and (b) gray-level plot along the diameter indicated as a dashed horizontal line in (a). (c) Enlargement of the gray level (circles) in the vicinity of the leftmost edge [highlighted by an ellipse in (b)] and hyperbolic tangent fit (solid line) for the determination of the edge position. The scale bar in (a) corresponds to $8\mu \mathrm{m}$.

Figure 6

Scattering of the incident plane wave ${\mathbf{E}}_{\mathrm{in}}(\mathbf{r})$ (wavelength ${\lambda }_1={\lambda }_0/n_1$) by a dielectric microsphere centered at the origin of the coordinate system. The scattered field ${\mathbf{E}}_s^{(1)}(\mathbf{r})$ is expanded as a superposition of plane waves. Each plane-wave component propagates along a direction defined by the scattering angle $\alpha$ (relative to the forward $z$ direction).