Three-Dimensional Polarization Control in Microscopy

We propose an approach to optical microscopy that enables full control over the three-dimensional polarization vector at the focal spot of a high-numerical-aperture lens. The input field to the lens is linearly polarized and no polarization optics are needed. This technique utilizes the azimuthal spatial degree of freedom of the input field. We find that only a small set of low-order azimuthal spatial harmonics contributes to the focused field on axis, and a simple transformation exists between the linear vector space of these harmonics and the three-dimensional polarization-vector space. Controlling the relative complex weights of these azimuthal harmonics produces any desired three-dimensional state of polarization.

Figure 1

(a) Schematic of the focusing geometry showing the input plane and an observation point in the vicinity of the focal point. (b),(c) Examples of input-field distributions designed to produce desired 3D states of polarization at the focal spot: amplitude and phase distributions of the $x$-polarized input field needed to produce (b) 45° linear polarization in the $x\mathrm{\text{−}}z$ plane, and (c) elliptical polarization in the $y\mathrm{\text{−}}z$ plane.

Figure 2

(a) An example of a 3D state of polarization and its projections on three orthogonal planes. (b) The amplitude and phase distributions of the input field required to produce, at the focal spot, the polarization state shown in (a).

Figure 3

(a) Intensity distributions for on-axis, off-axis, and total components of the $x$, $y$, and $z$ polarizations in the focal plane ($x\mathrm{\text{−}}y$) that correspond to an on-axis field having $E_x=E_y=E_z$ (see text for details). The scale bars refer to the last column normalized with respect to the peak of the total intensity distribution. All plots have a radius of $1\mu \mathrm{m}$. (b) Intensity distributions for on-axis, off-axis, and total components in the $y\mathrm{\text{−}}z$ plane for the total polarization. The dimensions of each square are $2\mu \mathrm{m}$ in the $y$ direction, as in (a), and $4\mu \mathrm{m}$ in the $z$ direction.