Since about 2008, my research has centered on highly focused fields, particularly the sort of field that results from a beam passing through a high numerical aperture (NA) lens system. This type of field is present in many systems of current interest, including (but not limited to) optical tweezers, confocal microscopes, photolithography, and laser-confined inertial fusion. Starting in 2011-2012 at the College of Wooster, I worked with a student to determine the feasibility of making measurements of the light scattered from micron-sized spherical particles in a modified version of the "Faster, Cheaper, Safer" optical tweezer proposed in [1]. At Elmhurst College, students in the Keystone program built a functioning optical tweezer over the summer, with modifications that will make it easier to obtain scattering measurements compared to the set-up used at the College of Wooster. Currently, a senior student is working to create a spatial filter to block the incident light from reaching the final camera, which will allow for only the scattered light to be recorded. I expect that he will have preliminary data before the end of the year.

This work was inspired by fantastic talk by Dr. Zach Smith about his work on the integrated Raman- and angular-scattering microscope [2], during which Miguel Alonso and I simultaneously realized that we could use CF fields to make the calculations much easier. It turns out that making Mie scattering calculations for a CF field (scalar or EM) is much simpler than other alternatives for Mie scattering of beams, due to a particularly simple form of the expansion coefficients. This is true regardless of the relative location of the center of the beam to the center of the scattering particle [3].

Complex-focus (CF) fields are a class of field that varies smoothly from a spherical wave (both converging and diverging) to a field that is a good approximation of a Gaussian beam with a parameter *kq*.
These fields closely resemble the complex-source point fields proposed in by Kravtsov [4] and Deschamps [5], but are analytic everywhere due to the presence of a sink at the same location as noted by Berry [6] and Sheppard and Saghafi [7].
To the left is a movie I generated in *Mathematica* showing a CF field as the parameter *kq* varies from 0.00 to 9.00. The horizontal axis is *kz* and the vertical axis is *kx* and the beam propagates in the *z* direction. The fields are a simple but very useful for a nonparaxial Gaussian beam. I have put CF fields to good use in everything I have published so far (except for the paper about my undergraduate research).

CF fields can also be generalized to electromagnetic fields as well (both with linear polarization [8] and radial and azimuthal polarizations [9]). What are radial and azimuthal polarization? They are novel polarization states that are mutually orthogonal. In a radially or azimuthally polarized field, the polarization vector points in a different direction at different points in the field. For radial polarization, the polarization vector points radially outward. Azimuthal polarization consists of the polarization vector pointing azimuthally (perpendicular to the radial direction).

Below, the movie shows a scatterer in a field with azimuthal polarization. Part (a) shows the incident field, (b) the total field (incident+scattered+inside the sphere), (c) the size of the coefficients needed (to check convergence with the number of terms used), and (d) far-field radiant intensity with backscattering in the center. Please use the link on the CV page to read the article [3] in full (Optics Express is open access).

It turns out that there are distinct differences in the far-field scattering patterns with incident polarization. These differences may be useful for increasing the speed and/or accuracy of Mie-scattering-based size measurements.

Since the Mie scattering work, I have "focused" on using orthogonal bases for computationally efficient wave propagation. The basic idea is that summing over basis elements is a lot quicker than computing double oscillatory integrals at every point of interest. However, existing bases that are a solution to the Helmholtz equation (such as the spherical harmonics) will not converge quickly for a high NA field because of their backward traveling components that are not present in the focused field. In [10] and [11], orthogonal scalar and EM bases that vary with a parameter like the complex-focus fields are proposed and simple fields are modeled. Then, in [12] these bases are used to model the propagation of partially-coherent fields (which really increases the time savings--now it's a double sum versus a quadruple oscillatory integral). Currently, I am investigating the interaction of these basis elements with a scattering particle and a paper describing the scalar case has recently been submitted for review.

[1] J. Bechhoefer and S. Wilson, Am. J. Phys. **70**, 393 (2002).

[2] Z.J. Smith and A.J. Berger, Opt. Lett. **43**, 714 (2008).

[3] N.J. Moore and M.A. Alonso, Opt. Express **16**, 5926 (2008).

[4] Yu. Kravtsov, Radiophys. and Quant. Elec. **10**, 719 (1967).

[5] G.A. Deschamps, Electron Lett. **7** , 684 (1971).

[6] M.V. Berry, J. Phys. A: Math. Gen. **27**, L391 (1994).

[7] C.J.R. Sheppard and S. Saghafi, Phys. Rev. A **57**, 2971 (1998).

[8] C.J.R. Sheppard and S. Saghafi, J. Opt. Soc. Am. A **16**, 1381 (1999).

[9] C.J.R. Sheppard and S. Saghafi, Opt. Lett. **24**, 1543 (1999).

[10] N.J. Moore and M.A. Alonso, J. Opt. Soc. Am. A **26**, 1754 (2009).

[11] N.J. Moore and M.A. Alonso, J. Opt. Soc. Am. A **26**, 2211 (2009).

[12] J.C. Petruccelli, N.J. Moore and M.A. Alonso, Opt. Commun. **283**, 4457 (2010).