The fabrication of high-performance spherical gradient refractive index (S-GRIN) optics requires nondestructive metrology techniques to inspect the samples. We have developed an angular-scan, swept-source-based, Fourier-domain optical coherence tomography (OCT) system centered at 1318 nm with 5 mm imaging depth capable of 180° polar scan and 360° azimuthal scan to investigate polymeric S-GRIN preforms. We demonstrate a method that enables simultaneous mapping of the group optical thickness, physical thickness, the radially-averaged group refractive index, and the transmitted wavefront of the S-GRIN preforms. The angular scan OCT imaging and metrology enables direct visualization, molding uniformity characterization, and optical property evaluations of the preforms. The results on two generations of S-GRIN preforms are discussed that showcase the evolution of the manufacturing process in response to the OCT metrology feedback.
© 2015 Optical Society of America
In recent years, there has been an increasing interest in exploring novel gradient refractive index (GRIN) optical materials to develop compact, lightweight, robust optical systems. Among the efforts to advance the GRIN materials, spherical GRIN (S-GRIN) lenses whose GRIN profile is symmetric about a point and therefore compositions of the same refractive index reside on spheres  have drawn notable attention due to their remarkable prospects for lightweight broadband management of light. S-GRIN components have demonstrated utility in regimes such as laser beam shaping , high-efficiency solar concentrators , and achromatization with S-GRIN singlets  enabling lightweight spaceborne optics and new classes of surgical loupes. A significant breakthrough in manufacturing has been made by the introduction of nanolayered-polymer-film-based S-GRIN optical components with an unparalleled control of their internal refractive index distribution . Among a series of optics produced using the S-GRIN manufacturing chain, dome-shaped meniscus S-GRIN preforms with ideally concentric surfaces that are thermoformed from flat axial GRIN sheets serve as the precursors to the final S-GRIN lenses. The S-GRIN profiles of the final lenses are critically dependent on those of the preforms that they were cut from. Thus, a nondestructive precision metrology tool is in strong demand to diagnose the defects of the preforms before the parts gain value in the manufacturing chain. Optical coherence tomography (OCT) is promising in fulfilling such a role in metrology. OCT is a technique for high-resolution, high-sensitivity, high-speed 3D imaging of samples, particularly after the introduction of Fourier domain OCT [6,7]. Originally developed as a powerful non-contact, non-invasive optical imaging tool in the field of biomedicine , OCT usage has recently expanded to be applied to metrology of materials. The capability of OCT for noninvasive measurement of the GRIN refractive index profile of a crystalline lens has been demonstrated by Verma et al.  and de Castro et al. [10,11]. OCT systems with linear scans have been reported to successfully provide nondestructive high-resolution three-dimensional visualization and quantitative evaluation of the internal structure of optically near-transparent flat films and sheets .
In the application of optical metrology of parts with high curvature as in the case of S-GRIN preforms, it is strategic for the OCT probe axis to be normal to the surface of a lens or preform for two reasons: (1) to eliminate the need to account for refraction at the first interface, and (2) to maximize the signal-to-noise ratio (SNR) in the OCT data set acquisition throughout the part, which is highly leveraged for precision metrology. The challenge then is to implement a scanning geometry that remains normal to the lens or preform surface for components with significant surface curvature. To remedy this issue, we developed and implemented an angular scan OCT system. This geometry allows for normal or near normal beam incidence on the entire imaging field for curved preform or lens surfaces and the high SNR images support precision metrology.
In this paper, we demonstrate the implementation of an angular scan OCT system we developed and discuss its imaging and metrology results on S-GRIN preforms. We first review the manufacturing process of the S-GRIN optics and provide the descriptions of the two preforms tested by OCT in Section 2; the layout and the key components of the angular scan OCT system are described in Sections 3; a method providing simultaneous measurements of the physical thickness and the radially-averaged refractive index of the preforms is described in Section 4; in Section 5, we demonstrate the image rendering and provide the optical thickness, physical thickness, index mapping, and transmitted wavefront metrology results on the two S-GRIN preforms; finally, we conclude the paper in Section 6.
2. Fabrication of S-GRIN polymeric optics and the preform specifications
The S-GRIN preforms investigated in this paper resulted from a unique fabrication chain for polymeric S-GRIN optics developed at Case Western Reserve University [13,14]. The details of the fabrication process were reported in prior work by the authors . To summarize, the manufacturing process starts with the fabrication of 50 µm thick nanolayered polymer films whose refractive indices are controlled, based on the effective medium theory  predictions, by tuning the volumetric ratio of the constituent polymer raw materials with substantially different refractive indices prior to film co-extrusion. The nanolayered films of varying refractive indices are then stacked based on the prescribed gradient index recipe and then thermo-compressed to create a consolidated axial GRIN sheet. This GRIN sheet is further compression-molded into a curved S-GRIN preform and subsequently diamond-turned into a final S-GRIN lens.
We previously reported on the nondestructive metrology of the nanolayered films and GRIN sheets performed by OCT . With an angular scan OCT system, whose concept was first illustrated in , but whose final setup and associated methodology are first reported here for preform metrology, the nondestructive inspection capability of OCT was further extended to curved S-GRIN samples. Two preforms, manufactured a few months apart, were examined by the angular scan OCT. The second preform illustrates clearly the successful evolution of the preform component manufacturing methods.
Figure 1 shows a photograph of the first S-GRIN preform and its specifications. Based on its design specifications, the first preform was a 3.2 mm ± 15 µm thick meniscus lens with 20 mm circular clear aperture and nominally concentric outer and inner surfaces whose nominal radii of curvature were 20.1 mm and 16.9 mm, respectively, with ± 0.6% radii tolerances. The preform was compressed and contoured from a 4.6 mm thick flat GRIN sheet through thermal-molding. The composition of the constituent 128 film layers of the preform was linearly varied every 100 to 125 microns based on the prescribed GRIN distribution recipe that varied between 74%/26% and 14%/86% PMMA/SAN17 from the exterior to the interior layers. The corresponding group refractive index was theoretically estimated to increase from 1.5087 to 1.5554 at 1318 nm across the total thickness of the preform as shown in Fig. 1(a). The group index rather than the more commonly known phase index is used in this paper as OCT directly measures the group optical path difference (OPD), a product of the group index and physical thickness, between sample and reference back-reflections due to the low coherence interferometry nature of OCT. The group index is associated with the phase index by a dispersion relation, whereis the dispersion slope at an arbitrary wavelength . The group index estimation for the preform at 1318 nm was based on the measured phase refractive indices of the library of constituent films at 632.8 nm at the manufacturing site and the dispersion characteristic of the co-extruded materials. The characterization of a sample with a discrete GRIN profile that displays discontinuities in index of refraction is advantageous for OCT.
A second S-GRIN preform was manufactured with thermoforming process refinements to the fabrication process adopted by the manufacturer. As has been previously reported , an issue due to the under-compression of the edge layers was discovered by an SS-OCT system applied to imaging the GRIN sheets, precursors to the preforms. In light of this undesirable thermo-compression effect, the stacking recipe of the film layers was modified to add extra film layers on both sides of the original 128 layers in advance of thermo-forming a GRIN sheet. The additional edge layers were then removed during the final diamond-turning of the preform into a plano-convex lens. Thus, the number of film layers within the new preform was increased to 154, which led to an increased nominal thickness of 3.84 mm ± 15 µm. The corresponding estimated group refractive index gradient at 1318 nm as a function of the depth of the preform is plotted in Fig. 2(a). A photograph and the size of the second preform are shown in Fig. 2(b) and 2(c).
3. Angular scan OCT system layout
3.1 OCT system description
The layout of the angular scan OCT system is shown schematically in Fig. 3. It is a swept-source-based Fourier domain OCT [17–19] built on a fiber-based Mach-Zehnder interferometer (MZI) configuration. The light source is a frequency swept laser (HSL-2100-WR, Santec, Japan) centered at 1318 nm with a full width at half maximum (FWHM) bandwidth of 125 nm. The axial point-spread-function of the system is approximately 10 µm (FWHM) in air and the effective frequency sweep rate of the light source is 20 kHz. 10% of the power input to the main OCT interferometer is split into the reference arm where a custom Fourier domain optical delay line is used for dispersion compensation ; the other 90% enters the sample arm and is directed by a specifically devised angular scanning configuration to perform high-accuracy distortion-free scan of a non-planar sample along the curvature of its first surface. The time-encoded back-reflected spectral interference signal is detected by a balanced photo-detector (1817-FC, New Focus, CA, USA), and then digitized on one channel of a two-channel, high-speed, 12-bit-resolution analog-to-digital converter (ADC) operating at 200 Msamples/s (NI PCI 5124, National Instruments, TX, USA).
The detected signal requires recalibration to the linear frequency space prior to Fourier transform, which is performed by using the time-frequency relation measured by an additional side MZI denoted by the dashed blue box in Fig. 3. Simultaneously with the detection of the main interference signal, the calibration signal is detected by a second balanced photo-detector and then digitized on another 8-bit-resolution ADC operating at 250 Msamples/s (NI PCI 5114, National Instruments, TX, USA). By performing a Fourier transform of a single recalibrated interference spectrum, an entire component reflectivity profile along the incident sample beam path is captured. By applying 2D scans of the incident beam, 3D imaging of the sample is achieved. The maximum sensitivity of the OCT system was measured to be 112 dB. The single-sided imaging depth range is about 5 mm as determined by a −10 dB sensitivity fall-off, which can be doubled in the full-range imaging .
3.2 Angular scan instrumentation and imaging
The angular scan OCT system is highlighted by the construction of its sample arm, as shown in Fig. 4, enabling angular scans of a spherical sample. The key components to this implementation are two motorized rotation stages actuating the motion translations along the polar (θ) and azimuthal (ϕ) dimensions, respectively and, when combined, enabling 3D scans in a spherical coordinate system. From the figure, the vertically standing ring-shaped rotation stage (RV240, Newport, CA, USA) provides steady carriage and smooth θ-rotation of a cage-based scanning head positioned parallel to the radius of the rotation stage, which provides precise axial alignment of the imaging optics. Mounted on the cage rails of the scanning head are a collimator lens that couples the light from the fiber to a 3 mm diameter collimated beam in free space, and an achromatic objective with 20 μm lateral resolution that can travel along the axis of the cage system to flexibly focus on samples of various radii and thicknesses.
In addition to the θ-rotation of the scanning head, the ϕ-scan component of the 3D angular scan is established by rotating the sample to be imaged azimuthally using a compact motorized rotary sample platform (CR1-Z7, Thorlabs, NJ, USA). For this integrated spherical coordinate scanning system, specific attention needs to be paid towards ensuring the alignment of the centers of rotation of both the θ and ϕ axes. As shown in Fig. 4, the ϕ-rotation platform is mounted between a set of base XYZ stages and a set of compact XY stages on the top that serves as the sample platform. The combination of the two sets of linear translation stages and the rotatory platform was essential to implementing the continuous rotationally symmetric normal scan of the spherical samples under test on a spherical grid. The base XY stages enabled the sample beam to be directed to the center of rotation of the ϕ-rotation platform, whereas the piezoelectric motor driven XY stages above allowed the center of curvature of the spherical sample to align axially with the center of rotation of the ϕ-rotation platform; the vertical Z stage was utilized to further adjust the height of the sample so that its spherical surfaces of interest and the angular motion trajectory of the θ-scanning head were concentric. Under the concentric alignment condition, a constant OPD between the reference and the spherical sample surface is observed when imaging during a rotational scan. A high grade roundness standard was used for calibrating the alignment of the angular scan system; various types of departures from a constant OPD during a 3D scan are correlated with distinct types of misalignment, which are removed during calibration.
During the experiment of imaging a preform, its outer surface was aligned to be concentric with the angular trajectory of the θ-rotation stage before imaging. The OCT system was then operated at sampling resolutions of 0.2 ̊ for both the polar (θ) and azimuthal (ϕ) dimensions and ~2 µm for the radial (r) dimension; a 3D data set consisting of 300 (θ) X 900 (ϕ) X 2000 (r) samples were collected to cover ± 30 ̊ polar scan and 180 ̊ azimuthal scan. The imaging covered the 20 mm clear aperture of the preform laterally and its full thickness radially. The raw 3D-matrix-based angular imaging data set was originally acquired in spherical coordinates, as illustrated in Fig. 5(b), where a sequence of 2D slices of images, each representing a depth (r) scan column-wise and a polar (θ) scan row-wise, were stacked along the third dimension of the azimuthal angles they had been taken at, as shown in Fig. 5(a). To facilitate direct visual correlation with the actual shape of the preform, further transformation from polar to Cartesian coordinates was performed in Matlab (R2013a, Mathworks) on the 2D slices of cross-sectional images to recover the curvature of the preform to the actual spatial proportion as shown in Fig. 5(c).
4. Simultaneous estimation of the average group refractive index and the physical thickness of S-GRIN preforms
The FD-OCT imaging directly provides depth sectioning of a sample based on the group OPD between the sample and reference arms without translating the sample. The measured OPD involves both the physical thickness and the group index that are desired to be extracted separately. Several methods have been proposed using low coherence interferometry to decouple the measurements of the refractive index and the physical thickness [21–24]. Based on the experimental configurations such as the integration with confocal scanning or multi-photon microscopy, the index profile of layered samples can be measured by tracking the focal shift from one surface to the next and combining with the optical path length knowledge. Uhlhorn et al. proposed a method of assessing the distortion induced on a plane reference by a crystalline lens inserted in the optical path to derive its average index and thickness . The method developed did not account for refraction induced by the curved outer surface of the crystalline lens and therefore Uhlhorn et al. limited the measurement to a single A-scan perpendicular to the apex of the lens. As such, the induced curvature of the plane located below the lens was solely qualitatively observed .
Here, we propose an approach based on detecting the optical path shift without requiring high cost microscope objectives or confocal scans of the sample, to estimate the 2D refractive index map of an S-GRIN preform averaged radially, which can be expressed as
Figure 6(a) shows a schematic cross-sectional view of the experimental sample setup in the angular-scan sample arm. This compact optomechanical cage is placed on top of the sample platform described in Section 3. From the figure, the preform is placed above and concentric to a calibration sphere, which is a grade 5 ball standard with a radius of 12.7 mm ± 20 nm and surface roughness < 5 nm (CaliBall, Optical Perspectives Group, AZ, USA). The calibration sphere is secured by a plastic retaining ring onto the lower cage mount with a compact Z micrometer for fine, individual adjustment of the height of the sphere. The upper cage mount lifted by 4 cage rods arranged in 30 mm square holds the preform and with its XY translators allows for lateral adjustments of the preform without perturbing the calibration sphere. A 26 mm-diameter clearance hole in the mount center avoids any beam obscuration during the ± 30-degree angular scan of samples. The alignment of both the preform and the calibration sphere consists of three steps. Initially, the optimal height of the preform was found first by adjusting the Z stage of the supporting sample platform until a straight line was observed for the outer surface of the preform during 3D angular scan (XY misalignment may be seen as a tilt in the line and would be removed during the final stage of alignment). Second, the preform was removed and the calibration sphere was aligned to be concentric with the trajectory of the scanning head by the Z-micrometer of the lower cage mount and the XY stages of the sample platform until a constant OPD for the sphere was observed during 3D scan. Finally, the preform was added back and aligned by adjusting only the XY translators of the upper cage mount until the outer surface of the preform was concentric with the trajectory of the scanning head. Note that the first stage of alignment in establishing the correct height of the preform was necessary as no further adjustment to the height of the preform could be made after the calibration sphere was aligned.
With the sample being aligned, the trajectory of the θ-scanner was ensured to be concentric with both the preform and the calibration sphere such that the incident rays passed through the preform without bending and were subsequently reflected along the normal of the calibration sphere. Our method thus circumvents the need to account for refraction by scanning the beam perpendicular to the outer surface of the preform and any departure from a concentric preform can be readily estimated throughout the preform effective aperture. Figure 6(b) shows a sequence of raw cross-sectional OCT images stacked along the azimuthal () angles. The optical thickness of the preform is obtained by computing the difference between the axial positions of the intensity peaks corresponding to the outer and inner surfaces, which can be expressed as
Subsequently, the preform was removed and only the calibration sphere was imaged again. A sequence of raw cross-sectional OCT images is shown in Fig. 6(c). As a result of the previous path of the preform being replaced by an air gap, the axial position of the calibration sphere on the OCT image is shifted upward by due to its reduced optical path difference relative to the reference arm, which can be expressed as26,27], despite the group refractive index being employed in OCT. It can be seen from Eqs. (2) and (3) that by measuring both the and the material parameters and can be obtained simultaneously as
5.1 Imaging and visualization of S-GRIN preforms
The two S-GRIN preforms described in Section 2 were imaged by the angular scan OCT following their manufacture. Figures 7(a) and 7(c) are two raw cross-sectional images of the first and the second preforms, respectively. The images were directly acquired by OCT where the top and bottom lines representing the outer and inner surfaces of the preform are denoted. The curvature of the inner surface and the significant deviation of the two surfaces from parallelism in Fig. 7(a), clearly seen visually, points to an optical thickness non-uniformity of the first preform that may be attributable to physical thickness variations caused by longitudinal (i.e., resulting in curvature of inner surface) or lateral (i.e., resulting in wedge in the thickness) decenters of the inner surface with respect to the outer surface or the GRIN non-uniformity angularly that may contribute both curvature and wedge depending on its type. Further determination requires using the method we described in Section 4 to decouple the various sources of error, the detailed results of which are presented in Section 5.2. Metrology results indicate that physical thickness errors were dominated by some decenter of the two surfaces, and this finding established by OCT prompted the manufacturer to seek an approach to examine and more accurately align the convex and concave molds. The second preform was made with a new pair of molds of improved figure quality and alignment tolerances. Its raw cross-sectional image shown in Fig. 7(c) exhibits much improved concentricity. Figures 7(b) and 7(d) are remapped from the raw images to the correct scale in Cartesian coordinates. In addition to the molding concentricity correction, the second preform also shows much reduced back-reflections from its internal structure and therefore significantly improved transmission. This is also a result of an earlier OCT inspection that revealed the undesirable layer structure within the earlier samples such as the first preform and led to a targeted manufacturing transformation .
5.2 Thickness, average index, and transmitted wavefront mapping of S-GRIN preforms
To inspect the possible defects of thickness variations or GRIN non-uniformity, 3D angular scan OCT imaging based on the method described in Section 4 was performed on each preform with a calibration sphere, followed by the imaging of the calibration sphere alone. The group optical thickness map of each preform and a measurement of the distortion of the surface profile of the calibration sphere caused by the preform were directly obtained from the 3D imaging (see Figs. 6(b) and 6(c)). The distortion of the calibration sphere is regarded as equivalent to the transmitted wavefront through the preform when light travels perpendicular to its outer surface. From these two maps, 2D mapping of the physical thickness and the radially-averaged group refractive index were further extracted based on Eqs. (4) and (5), respectively. All measurements were taken along the radial lines of the preform by setting the azimuthal and polar sampling intervals at 0.2 ̊.
The wedge and power in the thickness indicate a lateral shift and a longitudinal decenter of the inner surface with respect to the outer surface, which can be quantitatively evaluated based on the following equations, assuming and are much smaller than the preform radii of curvatures,
The results of the first preform are shown in Fig. 8. In agreement with the visualization of the cross-sectional images in Section 5.1, the first preform exhibits a significant optical thickness increase from the center to the edge that appears as a power term in Fig. 8(a). Also apparent in this plot is a tilt term (i.e., wedge in thickness). The physical thickness is further decoupled from the average group index and mapped out in Fig. 8(b) that preserves the power and tilt. By evaluating the magnitude of tilt and power, a lateral decenter of 81.5 ± 1.0 μm and a longitudinal decenter of 2.06 ± 0.02 mm were estimated based on Eqs. (6) and (8), respectively. The group refractive index of the preform at 1318 nm averaged radially is mapped out in Fig. 8(c). It was measured to be 1.5292 ± 0.0042 across the clear aperture of the preform, as compared to the theoretically predicted average group index of 1.5317. The measured transmitted wavefront of the preform is shown in Fig. 8(d), which is dominated by power and tilt. Due to the significant higher order aberrations in the part, 53% of its clear aperture was then masked to yield a negligible higher order residual (i.e., RMS < a quarter wave). A lower order 9-term Fringe Zernike fit was applied to the transmitted wavefront across the reduced aperture within the blue dashed circle in Fig. 8(d) for the evaluation of Seidel aberrations. The piston, tilt, and power terms were then removed and the remaining transmitted wavefront shown in Fig. 8(e) up to 14 ̊ polar angle was dominated by third order spherical aberration of 19.9 μm peak-to-valley (PV).
The results for the second preform are shown in Fig. 9. Figure 9(a) and 9(b) are the 2D maps of the group optical thickness and physical thickness of the preform across its 30 ̊ angular extent, respectively, which are dominated by tilt. The magnitude of tilt and power extracted from the physical thickness were estimated to correspond to a lateral decenter of 35.6 ± 0.5 μm and a longitudinal decenter of 110.3 ± 1.0 μm between the two surfaces of the preform based on Eqs. (6) and (7), respectively. The second preform shows more than an order of magnitude improvement over the first one in terms of the longitudinal alignment accuracy of the molds.
The group refractive index at 1318 nm averaged radially is mapped out in Fig. 9(c) and was measured to be 1.5319 ± 0.0017 across the clear aperture, as compared to the theoretically predicted average group index of 1.5318. The close agreement of the measured average index with the theoretical value to an accuracy of 10−4 and the field variation of the measured index on the order of 10−3 indicates a well-controlled S-GRIN profile of the second preform although some edge effects still remain.
From the OCT maps, distinct defects of the second preform at 3 o’clock and 5 o’clock are also revealed. They are consistent with the defects shown in the outer surface map of Fig. 9(g) measured by a Zygo GPI interferometer in reflection off the surface. From the Zygo outer surface measurement, results show that the residual error from a perfect spherical surface is dominated by astigmatism on the order of 5 waves PV at 632.8 nm. The inner surface shown in Fig. 9(f) is dominated by spherical aberration on the order of 1 wave PV. The OCT-measured transmitted wavefront of the preform across the clear aperture is shown in Fig. 9(d). To avoid the edge effect leading to higher order aberrations, the aperture was reduced by 13% to the area encircled by the blue dashed line in Fig. 9(d) for further Zernike fitting. The lower order 9-term Fringe Zernike fit of the transmitted wavefront up to 26 ̊ polar angle after removing piston, power and tilt is shown in Fig. 9(e), which has an RMS error of 2.4 μm. The residual is dominated by spherical aberration of 7.1 μm PV and astigmatism of 5.3 μm PV. The orientation of the astigmatism shown in the transmitted wavefront is slightly different from that of the outer surface, which indicates the dominating contribution from the internal GRIN to the overall wavefront aberrations. The impact of the outer surface astigmatism in transmission is a difference in the medium indices times the height, i.e., ± 0.8 um, which is significantly smaller than the astigmatism in the residual transmitted wavefront. The wavefront measurement is important to understand the combined effect of the surfaces and the homogeneity of the S-GRIN structure on wavefront aberrations. The measurement of the transmitted wavefront provided by OCT is readily obtained and is not susceptible to the spurious fringes that usually appear during the measurement of the transmitted wavefront from shells and windows carried out in conventional interferometric testing. By masking the edges of the transmitted wavefront and evaluating the residual aberrations, OCT metrology helps guide selecting the usable aperture of a preform for further diamond-turning into a lens.
From the OCT data sets, we also quantified the thickness variations of both preforms as a function of the polar angle (θ) as another means of evaluation of their concentricity. The results are plotted in Fig. 10. The azimuthally-averaged thicknesses of the first and the second preform increased by 199 μm (i.e., from 3.160 to 3.359 mm) and 16 μm (i.e., from 3.865 to 3.881 mm), respectively, from the center (θ = 0) to the periphery (θ = 30 ̊). The thicknesses of the first and the second preform along with their standard deviations measured by OCT across all 270,000 sampling points were 3.2525 mm ± 61.3 μm and 3.8701 mm ± 7.3 μm, respectively, which agreed with the micrometer measurements of 3.255 mm ± 57 μm and 3.871 mm ± 8 μm across 50 positions on each preform, considering the limited mechanical gauge precision.
In this paper, we presented the design and implementation of an angular scan OCT system optimized for the 3D characterization of curved S-GRIN samples, denoted as preforms. We demonstrated the high-resolution, high-SNR 3D imaging and nondestructive metrology capability on two generations of S-GRIN preforms. We proposed and implemented a method to simultaneously map the group optical thickness, physical thickness, the radially-averaged refractive index, and the transmitted wavefront of the preforms, so various errors in the preforms may be decoupled. Through the OCT inspections, significant improvement in the concentricity of the preforms and the conformity of the S-GRIN profile to the specifications was achieved, which showcases the capability of OCT in guiding manufacturing process iterations. In the future, the metrology of GRIN components may further leverage OCT to enable 3D refractive index profiling of the GRIN components.
Appendix: Estimation of the longitudinal and lateral decenters of preform surfaces
In this appendix, we describe the mathematical derivation for estimating the longitudinal and lateral decenters of the inner surface with respect to the outer surface of a preform from the thickness variations.
Figure 11 is a schematic of the cross section of a preform with its surfaces represented by their base spheres in the azimuthal plane (denoted as) determined by the centers of curvature of the outer and inner surfaces (and) and the vertex of the outer surface (). The radii of the outer and inner surfaces areand, respectively. The lateral and longitudinal shifts betweenandare denoted as and, respectively. and denote the thicknesses measured at a polar angle and along the vertex of the outer surface (i.e.,), respectively. A preform with ideally concentric surfaces would have a uniform thickness and overlapped and. Here, with the presence of an unknown due to the preform eccentricity, we estimate the amount of and from the measurements of in 2D. It is evident from the cross-sectional geometry that
Based on the Law of Cosines, the following relations hold for the triangles and, respectively, asEqs. (9)-(13), can be expressed asEq. (14) can be approximated from the Taylor expansion asEq. (15) can be rewritten asEquation (16) can be generalized to an arbitrarily given by projecting the lateral decenter to the new azimuthal angle as . Thus, in a general case, Eq. (16) becomesEq. (17) that, the lateral decenter leads to both linear () and third order () variations of , whereas the longitudinal decenterresults in a second order () variation. The above modeling is based on the assumption that the preform surfaces are spherical; any figure errors from the spheres may contribute to additional astigmatic, comatic and higher-order variations of (up to 36 Zernike terms being analyzed). Therefore, the magnitude of the linear tilt and second-order power components from the Zernike fit of are extracted to estimate and, respectively, as28].
Note that for the first preform we measured, when is not much smaller than, Eq. (14) may be approximated asEq. (20) yieldsEqs. (17)-(19), it can be derived that, in this case, is alternatively obtained (instead of using Eq. (19)) from the following equation
This work was supported by the Manufacturable Gradient Index Optics (M-GRIN) program of the Defense Advanced Research Projects Agency (DARPA) and the NYSTAR Foundation C050070. We thank Eric Baer and Kevin Thompson for stimulating discussions about this work. Finally, we thank the II-VI foundation for their support in OCT freeform metrology.
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