Kyle Fuerschbach

A new family of optical systems employing φ-polynomial surfaces

Unobscured optical systems have been in production since the 1960s. In each case, the unobscured system is an intrinsically rotationally symmetric optical system with an offset aperture stop, a biased input field, or both. This paper presents a new family of truly nonsymmetric optical systems that exploit a new fabrication degree of freedom enabled by the introduction of slow-servos to diamond machining; surfaces whose departure from a sphere varies both radially and azimuthally in the aperture. The benefit of this surface representation is demonstrated by designing a compact, long wave infrared (LWIR) reflective imager using nodal aberration theory. The resulting optical system operates at F/1.9 with a thirty millimeter pupil and a ten degree diagonal full field of view representing an order of magnitude increase in both speed and field area coverage when compared to the same design form with only conic mirror surfaces.

Theory of aberration fields for general optical systems with freeform surfaces

This paper utilizes the framework of nodal aberration theory to describe the aberration field behavior that emerges in optical systems with freeform optical surfaces, particularly φ-polynomial surfaces, including Zernike polynomial surfaces, that lie anywhere in the optical system. If the freeform surface is located at the stop or pupil, the net aberration contribution of the freeform surface is field constant. As the freeform optical surface is displaced longitudinally away from the stop or pupil of the optical system, the net aberration contribution becomes field dependent. It is demonstrated that there are no new aberration types when describing the aberration fields that arise with the introduction of freeform optical surfaces. Significantly it is shown that the aberration fields that emerge with the inclusion of freeform surfaces in an optical system are exactly those that have been described by nodal aberration theory for tilted and decentered optical systems. The key contribution here lies in establishing the field dependence and nodal behavior of each freeform term that is essential knowledge for effective application to optical system design. With this development, the nodes that are distributed throughout the field of view for each aberration type can be anticipated and targeted during optimization for the correction or control of the aberrations in an optical system with freeform surfaces. This work does not place any symmetry constraints on the optical system, which could be packaged in a fully three dimensional geometry, without fold mirrors.

Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces

This paper introduces the path forward for the integration of freeform optical surfaces, particularly those related to φ-polynomial surfaces, including Zernike polynomial surfaces, with nodal aberration theory. With this formalism, the performance of an optical system throughout the field of view can be anticipated analytically accounting for figure error, mount-induced errors, and misalignment. Previously, only misalignments had been described by nodal aberration theory, with the exception of one special case for figure error. As an example of these new results, three point mounting error that results in a Zernike trefoil deformation is studied for the secondary mirror of a two mirror and three mirror telescope. It is demonstrated that for the case of trefoil deformation applied to a surface not at the stop, there is the anticipated field constant contribution to elliptical coma (also called trefoil) as well as a newly identified field dependent contribution to astigmatism: field linear, field conjugate astigmatism. The magnitude of this astigmatic contribution varies linearly with the field of view; however, it has a unique variation in orientation with field that is described mathematically by a concept that is unique to nodal aberration theory known as the field conjugate vector.

Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors

We report on the assembly of an off-axis reflective imaging system employing freeform, φ-polynomial optical surfaces. The sensitivity of the system to manufacturing errors is studied for both a passive and active alignment approach. The as-built system maintains diffraction-limited performance in the long-wave infrared.

Interferometric measurement of a concave, φ-polynomial, Zernike mirror

We report on the surface figure measurement of a freeform, φ-polynomial (Zernike) mirror for use in an off-axis, reflective imaging system. The measurement utilizes an interferometric null configuration that is a combination of subsystems each addressing a specific aberration type, namely, spherical aberration, astigmatism, and coma.

A new generation of optical systems with φ-polynomial surfaces

Recent advances have made it viable to fabricate optical surfaces that are not rotationally symmetric using a new generation of diamond-turning machines. These surfaces can greatly extend the field of view of optical systems and provide compact solutions. Through the use of optimization and analysis methods that track aberration content throughout the field of view, two non-rotationally symmetric designs that provide a compact mirror based geometry with a 10 degree full field of view are presented and compared. It is shown that a φ-polynomial surface provides superior optical performance to an anamorphic asphere because the surface can be made comatic.

Designing with ϕ-polynomial surfaces

Recent advances have made it viable to fabricate optical surfaces that are not rotationally symmetric using a new generation of diamond-turning machines. This new fabrication capability allows for surfaces whose departure from a sphere varies both radially and azimuthally in the aperture to be machined into an optical surface. This new degree of freedom allows for the design of unobscured optical systems that are truly non-symmetric by tilting the optical surfaces themselves. With this new design degree of freedom, the aperture and field of view can be pushed to yield an order of magnitude increase in aerial coverage over current production while maintaining a compact solution. In one particular case, to be presented, a diffraction limited (< λ/10), long wave infrared (LWIR), F/1.9, 10° full field of view sensor telescope is designed by introducing these non-symmetric surfaces into the optical surface prescription.

Nodal aberration theory applied to freeform surfaces

When new three-dimensional packages are developed for imaging optical systems, the rotational symmetry of the optical system is often broken, changing its imaging behavior and making the optical performance worse. A method to restore the performance is to use freeform optical surfaces that compensate directly the aberrations introduced from tilting and decentering the optical surfaces. In order to effectively optimize the shape of a freeform surface to restore optical functionality, it is helpful to understand the aberration effect the surface may induce. Using nodal aberration theory the aberration fields induced by a freeform surface in an optical system are explored. These theoretical predications are experimentally validated with the design and implementation of an aberration generating telescope.

An analytic expression for the field dependence of FRINGE Zernike polynomial coefficients in optical systems that are rotationally nonsymmetric

Zernike polynomials have emerged as the preferred method of characterizing as-fabricated optical surfaces. From here, over time, they have come to be used as a sparsely sampled representation of the state of alignment of assembled optical systems both during and at the conclusion of the alignment process. We previously developed the field dependence that analytically interconnects the coefficients of the Zernike polynomial (which has to-date been characterized only by its aperture dependence) as a more complete representation of an aligned, rotationally symmetric optical system. Here, we extend this analytic expression for the RMS wavefront error to encompass the prediction of the performance of a misaligned optical system by expressing the field dependence within the framework of nodal aberration theory. This significant expansion to this valuable polynomial provides an important new tool for characterizing high performance optical systems throughout the optical design, fabrication, assembly, and interim and acceptance test process.

Airy Beams: Beyond Geometric Optics

Airy beams are a new class of nondiffracting beam predicted in 1979 and observed in 2007. Beam propagation methods are shown to be an effective way to study the behavior of the beams in propagation.