Experimental investigation in nodal aberration theory (NAT) with a customized Ritchey-Chrétien system: third-order coma.

Nodal aberration theory (NAT) describes the aberration properties of optical systems without symmetry. NAT was fully described mathematically and investigated through real-ray tracing software, but an experimental investigation is yet to be realized. In this study, a two-mirror Ritchey-Chrétien telescope was designed and built, including testing of the mirrors in null configurations, for experimental investigation of NAT. A feature of this custom telescope is a high-precision hexapod that controls the secondary mirror of the telescope to purposely introduce system misalignments and quantify the introduced aberrations interferometrically. A method was developed to capture interferograms for multiple points across the field of view without moving the interferometer. A simulation result of Fringe Zernike coma was generated and analyzed to provide a direct comparison with the experimental results. A statistical analysis of the measurements was conducted to assess residual differences between simulations and experimental results. The interferograms were consistent with the simulations, thus experimentally validating NAT for third-order coma.

Design, fabrication, and testing of convex reflective diffraction gratings.

The convex reflective diffraction grating is an essential optical component that lends itself to various applications. In this work, we first outline the design principles of convex diffraction gratings from wavefront quality and efficiency perspectives. We then describe a unique fabrication method that allows for the machining of convex diffraction gratings with variable groove structure, which is extendable to rotationally non-symmetric convex diffraction grating substrates. Finally, we demonstrate two quantitative wavefront measurement methods and respective experimental validation.

Freeform spectrometer enabling increased compactness.

We present optical designs with freeform optics in the context of hyperspectral imaging. Results show designs that are 5 × more compact in volume than similar designs using conventional spherical or aspherical surfaces. We will show how combining the concepts of spatial and spectral-band broadening, which will be introduced in this paper, led to the improvement in compactness that is uniquely enabled by freeform optics.

Volumetric rendering and metrology of spherical gradient refractive index lens imaged by angular scan optical coherence tomography system.

In this paper, we develop the methodology, including the refraction correction, geometrical thickness correction, coordinate transformation, and layer segmentation algorithms, for 3D rendering and metrology of a layered spherical gradient refractive index (S-GRIN) lens based on the imaging data collected by an angular scan optical coherence tomography (OCT) system. The 3D mapping and rendering enables direct 3D visualization and internal defect inspection of the lens. The metrology provides assessment of the surface geometry, the lens thickness, the radii of curvature of the internal layer interfaces, and the misalignment of the internal S-GRIN distribution with respect to the lens surface. The OCT metrology results identify the manufacturing defects, and enable targeted process development for optimizing the manufacturing parameters. The newly fabricated S-GRIN lenses show up to a 7x spherical aberration reduction that allows a significantly increased utilizable effective aperture.

MEMS-based handheld scanning probe with pre-shaped input signals for distortion-free images in Gabor-domain optical coherence microscopy.

High-speed scanning in optical coherence tomography (OCT) often comes with either compromises in image quality, the requirement for post-processing of the acquired images, or both. We report on distortion-free OCT volumetric imaging with a dual-axis micro-electro-mechanical system (MEMS)-based handheld imaging probe. In the context of an imaging probe with optics located between the 2D MEMS and the sample, we report in this paper on how pre-shaped open-loop input signals with tailored non-linear parts were implemented in a custom control board and, unlike the sinusoidal signals typically used for MEMS, achieved real-time distortion-free imaging without post-processing. The MEMS mirror was integrated into a compact, lightweight handheld probe. The MEMS scanner achieved a 12-fold reduction in volume and 17-fold reduction in weight over a previous dual-mirror galvanometer-based scanner. Distortion-free imaging with no post-processing with a Gabor-domain optical coherence microscope (GD-OCM) with 2 µm axial and lateral resolutions over a field of view of 1 × 1 mm2 is demonstrated experimentally through volumetric images of a regular microscopic structure, an excised human cornea, and in vivo human skin.

Ray-based optical design tool for freeform optics: coma full-field display.

The field of optical fabrication has progressed to a point where manufacturing optical quality freeform surfaces is no longer prohibitive. However, to stimulate the development of freeform systems, optical designers must be provided with the necessary tools. Full-field displays are an example of such a tool. Identifying the field dependence of the dominant aberrations of a freeform system is critical for a controlled optimization and with the help of full-field displays, this can be accomplished. Of specific interest is coma, an often system-limiting aberration and an aberration that has recently been directly addressed with freeform surfaces. In this research, we utilize nodal aberration theory to develop a ray-based method to generate a coma full-field display that circumvents wavefront fitting errors that can affect Zernike polynomial-based full-field displays for highly aberrated freeform starting designs.

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.

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.

Mobile device camera design with Q-type polynomials to achieve higher production yield.

The camera lenses that are built into the current generation of mobile devices are extremely stressed by the excessively tight packaging requirements, particularly the length. As a result, the aspheric departures and slopes on the lens surfaces, when designed with conventional power series based aspheres, are well beyond those encountered in most optical systems. When the as-manufactured performance is considered, the excessive aspheric slopes result in unusually high sensitivity to tilt and decenter and even despace resulting in unusually low manufacturing yield. Q(bfs) polynomials, a new formulation for nonspherical optical surfaces introduced by Forbes, not only build on orthogonal polynomials, but their unique normalization provides direct access to the RMS slope of the aspheric departure during optimization. Using surface shapes with this description in optimization results in equivalent performance with reduced alignment sensitivity and higher yield. As an additional approach to increasing yield, mechanically imposed external pivot points, introduced by Bottema, can be used as a design technique to further reduce alignment sensitivity and increase yield. In this paper, the Q-type polynomials and external pivot points were applied to a mobile device camera lens designed using an active RMS slope constraint that was then compared to a design developed using conventional power series surface descriptions. Results show that slope constrained Q-type polynomial description together with external pivot points lead directly to solutions with significantly higher manufacturing yield.

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.

Comparative assessment of freeform polynomials as optical surface descriptions.

Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enables the fabrication of freeform optics, or more specifically, optical surfaces for imaging applications that are not rotationally symmetric. Various forms of polynomials for describing freeform optical surfaces exist in optical design and to support fabrication. A popular method is to add orthogonal polynomials onto a conic section. In this paper, recently introduced gradient-orthogonal polynomials are investigated in a comparative manner with the widely known Zernike polynomials. In order to achieve numerical robustness when higher-order polynomials are required to describe freeform surfaces, recurrence relations are a key enabler. Results in this paper establish the equivalence of both polynomial sets in accurately describing freeform surfaces under stringent conditions. Quantifying the accuracy of these two freeform surface descriptions is a critical step in the future application of these tools in both advanced optical system design and optical fabrication.

Observation of the Gouy phase anomaly in astigmatic beams.

The Gouy phase anomaly, well established for stigmatic beams, is validated here for astigmatic beams. We simulate the predicted Gouy phase anomaly near astigmatic foci using a beam propagation algorithm integrated within lens design software. We then compare computational results with experimental data acquired using a modified Mertz-Sagnac interferometer. Both in simulation and in experiment, results show that a π/2-phase change occurs as the beam passes through each of the astigmatic foci, experimentally validating results derived in a recent paper by Visser and Wolf [Opt. Commun. 283, 3371-3375 (2010)].

Virtual skin biopsy with Gabor Domain optical coherence microscopy.

We report in-vivo volumetric optical coherence microscopy images of the skin, with resolution at the cellular level. With resolution of 2 µm both laterally and axially, structures below the skin as deep as 1 mm may be imaged at various anatomic locations. Custom optical instrumentation was designed, built, and integrated to achieve this unprecedented optical imaging resolution, in three dimensions, at clinically feasible configuration and speed.

Applying slope constrained Q-type aspheres to develop higher performance lenses.

It has recently been shown that the coefficients that specify the aspheric departure from a spherical surface in high NA lithographic lenses routinely require more significant digits than are available in even double precision computers when they are described as part of a power series in aperture-squared. The Q-type aspheric description has been introduced to solve this problem. An important by-product of this new surface description is that it allows the slope of a surface to be directly constrained during optimization. Results show that Q-type aspheric surfaces that are optimized with slope constraints are not only more testable, an original motivation, but, they can also lead to solutions that are less sensitive to assembly induced misalignments for lithographic quality lenses. Specifically, for a representative NA 0.75 lens, the sensitivity to tilt and decenter is reduced by more than 3X, resulting in significantly higher lens performance in-use.

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.

Cellular resolution optical coherence microscopy with high acquisition speed for in-vivo human skin volumetric imaging.

In this Letter, we report for the first time (to our knowledge) in-vivo volumetric optical coherence microscopy images of skin epidermal cells. We achieved micrometer-class resolution, 2 µm laterally and axially, with an acquisition speed of 23 K A-scans/s and over 90 dB sensitivity to a depth of 1 mm by employing a custom, liquid-lens-based, dynamic-focusing objective, a broadband light source, and a custom, astigmatism-corrected Czerny-Turner spectrometer with a high-speed complementary metal-oxide-semiconductor camera.

Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations.

Building on earlier work on the nodal aberration theory of third-order aberrations and a subset of fifth-order terms, this paper presents the multinodal field dependence of the family of aberrations describing the shape of the medial focal surface (the focal surface upon which the minimum RMS wavefront error is measured) and the astigmatic aberrations with respect to this surface through the fifth order. Specifically, the multinodal field dependence for W(420M) and W422 (the field-quartic medial surface and field-quartic astigmatism) are derived and presented as well as their influence on the magnitude and nodal field dependence of the companion lower-order terms, W(220M) and W222. This paper provides the first derivations of field-quartic aberrations presented by the author in the refereed literature.

Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces.

With the recent emergence of slow-servo diamond turning, optical designs with surfaces that are not intrinsically rotationally symmetric can be manufactured. In this paper, we demonstrate some important limitations to Zernike polynomial representation of optical surfaces in describing the evolving freeform surface descriptions that are effective for optical design and encountered during optical fabrication. Specifically, we show that the ray grids commonly used in sampling a freeform surface to form a database from which to perform a φ-polynomial fit is limiting the efficacy of computation. We show an edge-clustered fitting grid that effectively suppresses the edge ringing that arises as the polynomial adapts to the fully nonsymmetric features of the surface.

Airy beams: a geometric optics perspective.

Theoretically formulated in the 1970s within the context of nonrelativistic quantum mechanics, Airy beams have been experimentally realized for the first time only recently, paving the way to innovative optical techniques. While their remarkable features, a non-diffracting property and a transverse shift of the intensity maximum during propagation, are currently theoretically described from the wave optics viewpoint, here their exact relation to rays and geometric wavefront aberrations is revealed using a wavefront family that includes two-dimensional Airy beams. Several members of this family are computationally and experimentally implemented here. The lateral shift of Airy beams during propagation is presented in the context of the three-dimensional caustic representation. This new description allows re-emphasizing the use of "Airy-like" beams in computational imaging for depth of focus extension.