CeFO-Related Publications

@article{Ramirez-Andrade2024,

title = {Height reconstructions from geometric wavefronts using vision ray metrology},

author = {Ana Hiza Ramirez-Andrade and Konstantinos Falaggis},

url = {https://centerfreeformoptics.org/wp-content/uploads/2024/12/ao-63-33-8630-1-1.pdf},

doi = {https://doi.org/10.1364/AO.539226},

year  = {2024},

date = {2024-11-13},

urldate = {2024-11-13},

journal = {Applied Optics},

volume = {63},

number = {33},

pages = {8630-8640},

abstract = {A recently reported vision ray metrology technique [Opt. Express 29, 43480 (2021)] measures geometric wavefronts with high precision. This paper introduces a method to convert these wavefront data into height information, focusing on the impact of back surface flatness and telecentricity errors on measurement accuracy. Systematic errors from these factors significantly affect height measurements. Using ray trace simulations, we esti- mate reconstruction errors with various plano-concave and plano-convex elements. We also developed a calibration technique to mitigate telecentricity errors, achieving submicron accuracy in surface reconstruction. This study pro- vides practical insights into vision ray metrology systems, highlighting validity limits, emphasizing the importance of calibration for larger samples, and establishing system alignment tolerances. The reported technique for the conversion of geometric wavefronts to surface topography employs a direct non-iterative ray-tracing-free method. It is ideally suited for reference-free metrology with application to freeform optics manufacturing.},

keywords = {CeFO related, metrology},

pubstate = {published},

tppubtype = {article}

}

@article{Swain2024,

title = {Wavefront sensing with optical differentiation powered by deep learning},

author = {Biswa R. Swain, M. Akif Qadeer, Christophe Dorrer, Renuka Manjula Narayanan, Jannick P. Rolland, and Jie Qiao},

url = {https://centerfreeformoptics.org/wp-content/uploads/2024/12/ol-49-18-5216.pdf},

doi = {https://doi.org/10.1364/OL.530559},

year  = {2024},

date = {2024-09-11},

journal = {Optics Letter},

volume = {49},

number = {18},

pages = {5216-5219},

abstract = {We report the experimental demonstration of an optical differentiation wavefront sensor (ODWS) based on binary pixelated linear and nonlinear amplitude filtering in the far- field. We trained and tested a convolutional neural network that reconstructs the spatial phase map from nonlinear- filter-based ODWS data for which an analytic reconstruction algorithm is not available. It shows accurate zonal retrieval over different magnitudes of wavefronts and on randomly shaped wavefronts. This work paves the way for the imple- mentation of simultaneously sensitive, high dynamic range, and high-resolution wavefront sensing.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{M.2023,

title = {Sigma vector calculations in nodal aberration theory and experimental validation using a Cassegrain telescope},

author = {Yeşiltepe M., E. Arpa, M. Ekinci, O. Karci, J.P. Rolland},

doi = {10.1364/OE.502744},

isbn = {1094-4087},

year  = {2023},

date = {2023-12-01},

journal = {Optic Express},

volume = {31},

issue = {25},

pages = {42373},

abstract = {Nodal Aberration Theory (NAT) was developed to explain the field dependency of aberration field centers in the image plane of nominally rotationally symmetric optical systems that have lost their symmetry through misalignments. A new insight into the theory led to calculating the sigma vectors, which locate the aberration field centers, using the angle between a real-ray trace of the optical axis ray (OAR) and the normal of the local surface where “local” refers to the object and image optical spaces of that surface. Here, we detail the sigma vector calculations for general optical systems and provide an experimental investigation of a misaligned system with a high-precision customized Cassegrain telescope. In the simulations, a Newtonian telescope, a Cassegrain telescope, and a three-mirror anastigmat telescope were misaligned intentionally in ray-tracing software. The sigma vectors were calculated analytically for the third-order aberrations of astigmatism and coma. Experimentally, the same perturbations were implemented for the Cassegrain telescope system, and the aberrations were quantified through interferometric measurements on a grid of field points in the image plane that verified the analytical derivation and simulations.},

keywords = {CeFO related, metrology},

pubstate = {published},

tppubtype = {article}

}

@article{Yesiltepe2023,

title = {Sigma vector calculations in nodal aberration theory and experimental validation using a Cassegrain telescope},

author = {M. Yesiltepe and A. Bauer and O. Karci and J. P. Rolland},

doi = {https://doi.org/10.1364/OE.502744},

year  = {2023},

date = {2023-12-01},

urldate = {2023-12-01},

journal = {Optics Express},

volume = {31},

issue = {25},

pages = {42373-42387},

abstract = {Nodal Aberration Theory (NAT) was developed to explain the field dependency of aberration field centers in the image plane of nominally rotationally symmetric optical systems that have lost their symmetry through misalignments. A new insight into the theory led to calculating the sigma vectors, which locate the aberration field centers, using the angle between a real-ray trace of the optical axis ray (OAR) and the normal of the local surface where “local” refers to the object and image optical spaces of that surface. Here, we detail the sigma vector calculations for general optical systems and provide an experimental investigation of a misaligned system with a high-precision customized Cassegrain telescope. In the simulations, a Newtonian telescope, a Cassegrain telescope, and a three-mirror anastigmat telescope were misaligned intentionally in ray-tracing software. The sigma vectors were calculated analytically for the third-order aberrations of astigmatism and coma. Experimentally, the same perturbations were implemented for the Cassegrain telescope system, and the aberrations were quantified through interferometric measurements on a grid of field points in the image plane that verified the analytical derivation and simulations.},

keywords = {CeFO related, design},

pubstate = {published},

tppubtype = {article}

}

@article{Xiong2023,

title = {All-dielectric hybrid VIS-NIR dual-function metasurface},

author = {P. Xiong and D. K. Nikolov and F. Cheng and J. P. Rolland and A. N. Vamivakas },

url = {https://arxiv.org/abs/2310.09393},

doi = {https://doi.org/10.48550/arXiv.2310.09393},

year  = {2023},

date = {2023-10-13},

urldate = {2023-10-13},

journal = {ArXiv},

volume = {2310},

pages = {09393},

abstract = {Metasurfaces are a promising technology that can serve as a compact alternative to conventional optics while providing multiple functions depending on the properties of the incident light, such as the wavelength, polarization, and incident angle. Here, we demonstrate a hybrid VIS-NIR dielectriac metasurface that can reflect 940 nm light into a specified direction while transmitting visible light (450-750 nm). The dual functionality is realized by combining an aperiodic distributed Bragg reflector with dielectric meta-tokens. Experimental demonstration is also reported, showing an anomalous reflection of near-infrared (NIR) light within a 20o full field-of-view (FOV) and the transmission of wavelengths from 450 nm to 750 nm.},

keywords = {CeFO related, design},

pubstate = {published},

tppubtype = {article}

}

@article{Davies2023,

title = {Grinding of silicon carbide for optical surface fabrication. Part II. Subsurface damage},

author = {Prithiviraj Shanmugam and John C. Lambropoulos and Matthew A. Davies},

year  = {2023},

date = {2023-06-01},

urldate = {2023-06-01},

journal = {Applied Optics},

volume = {62},

issue = {14},

pages = {3778-3796},

abstract = {This paper is the second part of a study of the grinding of three different grades of silicon carbide (SiC) under the same conditions. In this paper, subsurface damage is analyzed using magnetorheological finishing (MRF). The MRF ribbon is brought into contact with the surface and allowed to dwell for different lengths of time to produce dimples or spots at different depths. The roughness parameters are evaluated at the base of the spots. As the spot depth increases the roughness parameters decrease, eventually saturating at a steady-state value. The depth at which saturation occurs is much greater than the initial peak-to-value roughness of the surface and is therefore assumed to be correlated to the depth of subsurface damage in the material. Estimates of the damage depth are comparable to other estimates given in the literature. The validity of this technique is assessed across different grades of SiC under different grinding conditions, and limitations are identified. The study shows that the microstructure of the SiC grade is a major factor that affects the validity of the technique. The technique is suitable for SiC grades that have a more homogeneous microstructure such as chemical vapor deposited or chemical vapor composite grades. The presence of porosity as in the direct sintered grade, or the presence of a secondary phase, for example, silicon in the reaction-bonded and silicon-infiltrated grades, could hinder the technique from providing conclusive results.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{Goodsell2023,

title = {Metagrating meets the geometry-based efficiency limit for AR waveguide in-couplers},

author = {J. Goodsell and P. Xiong and D. K. Nikolov and A. N. Vamivakas and J. P. Rolland},

doi = {https://doi.org/10.1364/OE.480092},

year  = {2023},

date = {2023-01-24},

urldate = {2023-01-24},

journal = {Optics Express},

volume = {31},

issue = {3},

pages = {4599-4614},

abstract = {Recently, augmented reality (AR) displays have attracted considerable attention due to the highly immersive and realistic viewer experience they can provide. One key challenge of AR displays is the fundamental trade-off between the extent of the field-of-view (FOV) and the size of the eyebox, set by the conservation of etendue sets this trade-off. Exit-pupil expansion (EPE) is one possible solution to this problem. However, it comes at the cost of distributing light over a larger area, decreasing the overall system's brightness. In this work, we show that the geometry of the waveguide and the in-coupler sets a fundamental limit on how efficient the combiner can be for a given FOV. This limit can be used as a tool for waveguide designers to benchmark the in-coupling efficiency of their in-coupler gratings. We design a metasurface-based grating (metagrating) and a commonly used SRG as in-couplers using the derived limit to guide optimization. We then compare the diffractive efficiencies of the two types of in-couplers to the theoretical efficiency limit. For our chosen waveguide geometry, the metagrating's 28% efficiency surpasses the SRG's 20% efficiency and nearly matches the geometry-based limit of 29% due to the superior angular response control of metasurfaces compared to SRGs. This work provides new insight into the efficiency limit of waveguide-based combiners and paves a novel path toward implementing metasurfaces in efficient waveguide AR displays.},

keywords = {CeFO related, design},

pubstate = {published},

tppubtype = {article}

}

@article{Davies2022,

title = {Grinding of silicon carbide for optical surface fabrication, Part 1: surface analysis},

author = {Prithiviraj Shanmugam and John C. Lambropoulos and Matthew A, Davies},

year  = {2022},

date = {2022-06-01},

urldate = {2022-06-01},

journal = {Applied Optics},

volume = {61},

issue = {15},

pages = {4579-4590},

abstract = {This paper presents a study of the grinding of three different grades of silicon carbide (SiC) under the same conditions. Surface topography is analyzed using coherent scanning interferometry and scanning electron microscopy. The study provides a baseline understanding of the process mechanics and targets effective selection of process parameters for grinding SiC optics with near optical level surface roughness, thus reducing the need for post-polishing. Samples are raster and spiral ground on conventional precision machines with metal and copper-resin bonded wheels under rough, medium, and finish grinding conditions. Material microstructure and grinding conditions affect attainable surface roughness. Local surface roughness of less than 3 nm RMS was attained in both chemical vapor deposition (CVD) and chemical vapor composite (CVC) SiC. The tool footprint is suitable for sub-aperture machining of a large freeform optics possibly without the need for surface finish correction by post-polishing. Subsurface damage will be assessed in Part 2 of this paper series.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{goodsell_2022,

title = {Method for minimizing lens breathing with one moving group},

author = {J. Goodsell and V. Blahnik and J. P. Rolland},

doi = {https://doi.org/10.1364/OE.457420},

year  = {2022},

date = {2022-05-23},

journal = {Optics Express},

volume = {30},

issue = {11},

pages = {19494-19509},

abstract = {Lens breathing in movie cameras is the change in the overall content of a scene while bringing subjects located at different depths into focus. This paper presents a method for minimizing lens breathing or changing angular field-of-view while maintaining perspective by moving only one lens group. To maintain perspective, the stop is placed in a fixed position where no elements between the scene and the stop can move, thus fixing the entrance pupil in one location relative to the object fields. The result is perspective invariance while refocusing the lens. Using paraxial optics, we solve for the moving group's position to focus on every object position and eliminate breathing between the minimum and maximum object distances. We investigate the solution space for optical systems with two positive groups or a positive and a negative group (i.e., retrofocus and telephoto systems). We explain how to apply this paraxial solution to existing systems to minimize breathing. The results for two systems altered using this method are presented. Breathing improved by two orders of magnitude in both cases, and performance specifications were still met when compared to the initial systems.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{Koerber22b,

title = {Reduction of retrace error from broadband variable transmission spheres in Fizeau interferometry},

author = {J. Koerber and G.D. Boreman and T.J. Suleski},

doi = {https://doi.org/10.1364/OPTCON.458860},

year  = {2022},

date = {2022-04-25},

journal = {Optics Continuum},

volume = {1},

number = {5},

pages = {1067-1076},

keywords = {CeFO related, metrology},

pubstate = {published},

tppubtype = {article}

}

@article{karci_2022,

title = {Experimental investigation in nodal aberration theory (NAT): separation of astigmatic figure error from misalignments in a Cassegrain telescope},

author = {O. Karci and M. Yesiltepe and E. Arpa and Y. Wu and M. Ekinci and J. P. Rolland},

doi = {https://doi.org/10.1364/OE.453788},

year  = {2022},

date = {2022-03-28},

journal = {Optics Express},

volume = {30},

issue = {7},

pages = {11150-11164},

abstract = {We present simulations and experimental validations for separating astigmatic figure error from misalignments in Nodal Aberration Theory (NAT) with a high-precision Cassegrain telescope. Both the primary mirror figure error and the secondary mirror misalignments induce binodal astigmatism for the telescope systems. The separation of these two aberration factors plays a crucial role in the telescope alignment process. In this study, the figure error of the aspheric primary mirror of the Cassegrain telescope induced by the mirror mounts was measured interferometrically utilizing a computer-generated hologram (CGH). According to the primary mirror figure error, the astigmatic node locations in the image plane were simulated using real raytracing. The center of the nodes was located on the field center, and the nodes were placed symmetrically with respect to the field center in the image plane. The telescope's alignment was performed using the simulation results, and the node locations were measured on a grid of field points interferometrically. Thereafter, secondary mirror misalignments around the coma-free pivot point were introduced into the optical system, and the node's center was shifted from the field center in the image plane as predicted by NAT. The simulations and interferometric field measurements were performed and compared on a grid of field points for the misaligned state in the presence of primary mirror figure error. The experimental results confirm the predictions from NAT. Statistical analysis was also performed to confirm the accuracy and stability of the measurements.},

keywords = {CeFO related},

pubstate = {published},

tppubtype = {article}

}

@article{Koerber22a,

title = {Broadband Variable Transmission Sphere for Fizeau Interferometry},

author = {J. Koerber and G.D. Boreman and T.J. Suleski},

doi = {https://doi.org/10.3390/opt3010011},

year  = {2022},

date = {2022-03-10},

journal = {Optics},

volume = {3},

pages = {88-98},

keywords = {CeFO related, metrology},

pubstate = {published},

tppubtype = {article}

}

@article{falaggis_2022,

title = {Freeform Optics: Introduction},

author = {K. Falaggis and J. P. Rolland and F. Duerr and A. Sohn},

doi = {https://doi.org/10.1364/OE.454788},

year  = {2022},

date = {2022-02-14},

journal = {Optics Express},

volume = {30},

issue = {4},

pages = {6450-6455},

abstract = {This feature issue of Optics Express highlights 28 state-of-the-art articles that capture a snapshot of the recent developments in the field of freeform optics. As an introduction, the editors provide an overview of all published articles, which cover a broad range of topics in freeform optics. The wide variety of applications presented here demonstrates that freeform optics is a growing and vibrant field with many more innovations to come.},

keywords = {CeFO related},

pubstate = {published},

tppubtype = {article}

}

@article{takaki_cub2022,

title = {Schemes for cubature over the unit disk found via numerical optimization},

author = {N. Takaki and G. W. Forbes and J.P. Rolland},

doi = {https://doi.org/10.1016/j.cam.2021.114076},

isbn = {0377-0427},

year  = {2022},

date = {2022-01-03},

journal = {Journal of Computational and Applied Mathematics},

volume = {407},

pages = {114076},

abstract = {Cubature schemes, in this case for uniformly weighted integration over the unit disk, enable exact evaluation of numerical integrals of polynomials but have been explicitly constructed for only low or moderate degrees. In this paper, cubature formulae are discovered for a wider range of degrees by leveraging numerical optimization. These results include a degree-17 cubature scheme with fewer points than existing solutions and up to a degree-77 solution with 1021 cubature points. Optimization heuristics and patterns in the distributions of cubature points are discussed, which serve as vital guides in this work. For example, these heuristics leverage a connection to circle-packing configurations to facilitate the discovery of fully symmetric cubature schemes.},

keywords = {CeFO related},

pubstate = {published},

tppubtype = {article}

}

@article{Hu2021,

title = {Morphology and waveguiding properties of ultrafast-laser-inscribed type-II waveguides in IG2},

author = {W. Hu and M. Kilinc and W. Gebremichael and C. Dorrer, and J. Qiao},

url = {https://opg.optica.org/ome/fulltext.cfm?uri=ome-12-1-360&id=466364},

year  = {2021},

date = {2021-12-24},

urldate = {2021-12-24},

journal = {Optical Materials Express},

volume = {12},

number = {1},

pages = {360-373},

abstract = {Ultrafast-laser-inscribed type-II waveguides inside the chalcogenide glass IG2 and the morphology of the inscribed tracks are presented for the first time to our knowledge. Self-organized periodic structures induced by the ultrafast laser in the tracks are characterized. The near-infrared waveguiding properties are studied for various pulse energies, durations, repetition rates, and polarization states, as well as different separations between the two written tracks. A propagation loss of ∼ 2.3 dB/cm is demonstrated in a type-II waveguide formed in IG2.},

keywords = {manufacturing, related},

pubstate = {published},

tppubtype = {article}

}

@article{Feng2021,

title = {Femtosecond-Laser-Ablation Dynamics in Silicon Revealed by Transient Reflectivity Change},

author = {T. Feng and G. Chen and H. Han and J. Qiao},

url = {https://www.mdpi.com/2072-666X/13/1/14#

https://centerfreeformoptics.org/wp-content/uploads/2023/02/micromachines-13-00014-v2.pdf},

doi = {https://doi.org/10.3390/mi13010014},

year  = {2021},

date = {2021-12-23},

urldate = {2021-12-23},

journal = {Micromachines},

volume = {13},

number = {1},

abstract = {The dynamics of ablation in monocrystalline silicon, from electron-hole plasma generation to material expansion, upon irradiation by a single femtosecond laser pulse (1030 nm, 300 fs pulse duration) at a wide range of fluences is investigated using a time-resolved microscopy technique. The reflectivity evolution obtained from dynamic images in combination with a theoretical Drude model and a Two-Temperature model provides new insights on material excitation and ablation process. For all fluences, the reflectivity increased to a temporary stable state after hundreds of femtoseconds. This behavior was predicted using a temperature-dependent refractive index in the Drude model. The increase in velocity of plasma generation with increasing fluence was theoretically predicted by the Two-Temperature model. Two ablation regimes at high fluences (>0.86 J/cm2) were identified through the measured transient reflectivity and ablation crater profile. The simulation shows that the fluence triggering the second ablation regime produces a boiling temperature (silicon, 2628 K).},

keywords = {manufacturing, related},

pubstate = {published},

tppubtype = {article}

}

@article{papa_OE21,

title = {Survey of the four-mirror freeform imager solution space},

author = {J. C. Papa and J. M. Howard and J. P. Rolland},

doi = {https://doi.org/10.1364/OE.442943},

year  = {2021},

date = {2021-12-06},

journal = {Optics Express},

volume = {29},

issue = {25},

pages = {41534-41551},

abstract = {We present the results and methods of a survey of the four-mirror freeform imager solution space. This survey is a grid search over the parameters that remain after reducing the solution space’s degrees of freedom using first-, second-, and third-order aberration theory. We then optimize the systems’ higher-order surface shapes through sixth-order. Finally, we discuss general trends in the systems’ performance versus volume and etendue and their utility in trade studies.},

keywords = {CeFO related},

pubstate = {published},

tppubtype = {article}

}

@article{Swain2021,

title = {Telephoto-lens-based Optical Differentiation Wavefront Sensor for freeform metrology},

author = {BR Swain and C. Dorrer and J. Qiao},

url = {https://opg.optica.org/oe/fulltext.cfm?uri=oe-29-23-38395&id=464483},

doi = {https://doi.org/10.1364/OE.443558},

year  = {2021},

date = {2021-11-03},

urldate = {2021-11-03},

journal = {Opt. Express},

volume = {29},

number = {23},

pages = {38395-38403},

abstract = {We report an Optical Differentiation Wavefront Sensor based on a telephoto lens system and binary pixelated filters. It provides a five-fold reduction in the system length compared to a 4f system with identical effective focal length. Measurements of phase plates with this system are compared to measurements performed with a commercial low-coherence interferometer. The telephoto-lens-based system can measure wavefronts with accuracy better than λ/10 Root Mean Squared (RMS) at λ=633 nm. Experimental investigation shows that the system has a high tolerance to components alignment errors.},

keywords = {metrology, related},

pubstate = {published},

tppubtype = {article}

}

@article{karci_NAT21,

title = {Experimental investigation of binodal astigmatism in nodal aberration theory (NAT) with a Cassegrain telescope system},

author = {O. Karci and E. Arpa and M. Ekinci and J. P. Rolland},

doi = {https://doi.org/10.1364/OE.427164},

year  = {2021},

date = {2021-06-21},

journal = {Optics Express},

volume = {29},

issue = {13},

pages = {19427-19440},

abstract = {We present simulations and an experimental investigation of binodal astigmatism in

nodal aberration theory (NAT) for a customized, high-precision Cassegrain telescope system. The

telescope system utilizes a five-axis, piezo-actuated flexural mechanism to introduce secondary

mirror misalignments and generate aberrations intentionally. The induced aberrations are

measured interferometrically and quantified for a grid of field points on the telescope system’s

image plane. For this purpose, a coma-free pivot point of the secondary mirror was simulated for

isolating the binodal astigmatism field response. The separation of the nodes is proportional to

the introduced misalignments. A simulation of Fringe Zernike coma and binodal astigmatism

was generated using a real ray trace model of the optical system and analyzed to compare to

the experimental results. A statistical analysis of the measurements was performed to show the

experimental results’ accuracy and stability. The experimental results were consistent with the

simulations, hence experimentally validating NAT for binodal astigmatism for the first time.

},

keywords = {CeFO related},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {Metaform optics: Bridging nanophotonics and freeform optics},

author = {Daniel K. Nikolov and Aaron Bauer and Fei Cheng and Hitoshi Kato and A. Nick Vamivakas and Jannick P. Rolland},

url = {https://www.science.org/doi/abs/10.1126/sciadv.abe5112},

doi = {10.1126/sciadv.abe5112},

year  = {2021},

date = {2021-04-30},

urldate = {2021-04-30},

journal = {Science Advances},

volume = {7},

number = {18},

pages = {eabe5112},

abstract = {The demand for high-resolution optical systems with a compact form factor, such as augmented reality displays, sensors, and mobile cameras, requires creating new optical component architectures. Advances in the design and fabrication of freeform optics and metasurfaces make them potential solutions to address the previous needs. Here, we introduce the concept of a metaform—an optical surface that integrates the combined benefits of a freeform optic and a metasurface into a single optical component. We experimentally realized a miniature imager using a metaform mirror. The mirror is fabricated via an enhanced electron beam lithography process on a freeform substrate. The design degrees of freedom enabled by a metaform will support a new generation of optical systems.},

keywords = {Aberration correction, augmented reality, CeFO manufacturing, CEFO metrology, freeform, head display, Image quality, Optical design, relay optics, rolland, tomography},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {Twyman effect in thin curved optics: effects of variable thickness and curvature},

author = {Sanjay N. Lakshmanan and John C. Lambropoulos},

year  = {2021},

date = {2021-02-19},

urldate = {2021-02-19},

journal = {Applied Optics},

volume = {60},

issue = {6},

pages = {1780-1789},

abstract = {Grinding or finishing of thin curved optics introduces surface-localized stresses that may result in bending deformations, thus affecting optical performance (Twyman effect). The Twyman effect depends on local thickness of the optic as well as its curvature. We use numerical and analytical techniques to assess the effects of these variables for shallow and deeply curved axisymmetric optics. A useful metric of comparison, valid for shallow optics, is the deformation of a similar flat optic (window). This comparison is inadequate for more curved optics, which demonstrate a smaller deformation that is now localized in a boundary layer near the optic-free edge. The effects of thickness variation are also presented. Normal deflection and slope for a positive optic (thickness at the center greater than the tip) are well approximated by a constant thickness optic. This is not the case for a negative optic, where deformations near the apex are more pronounced, thus extending significant deformations through the full optic aperture.},

keywords = {CeFO related, fabrication},

pubstate = {published},

tppubtype = {article}

}