Differentiation of breast tissue types for surgical margin assessment using machine learning and polarization-sensitive optical coherence tomography.

We report an automated differentiation model for classifying malignant tumor, fibro-adipose, and stroma in human breast tissues based on polarization-sensitive optical coherence tomography (PS-OCT). A total of 720 PS-OCT images from 72 sites of 41 patients with H&E histology-confirmed diagnoses as the gold standard were employed in this study. The differentiation model is trained by the features extracted from both one standard OCT-based metric (i.e., intensity) and four PS-OCT-based metrics (i.e., phase difference between two channels (PD), phase retardation (PR), local phase retardation (LPR), and degree of polarization uniformity (DOPU)). Further optimized by forward searching and validated by leave-one-site-out-cross-validation (LOSOCV) method, the best feature subset was acquired with the highest overall accuracy of 93.5% for the model. Furthermore, to show the superiority of our differentiation model based on PS-OCT images over standard OCT images, the best model trained by intensity-only features (usually obtained by standard OCT systems) was also obtained with an overall accuracy of 82.9%, demonstrating the significance of the polarization information in breast tissue differentiation. The high performance of our differentiation model suggests the potential of using PS-OCT for intraoperative human breast tissue differentiation during the surgical resection of breast cancer.

Spurious fringe processing for dielectric metasurface profile measurement using white-light scanning interferometry.

Dielectric metasurfaces, which are capable of manipulating incident light, have been a novel branch of flat optics. This modulation ability is realized by nanostructures with space-variant geometrical parameters such as height and diameter. Therefore, accurate profile measurement of metasurfaces is of great importance. White-light scanning interferometry is widely used for profile measurement. The step height is retrieved by locating the envelope's peak. However, spurious fringes attached to the desired fringes were observed at the measured area near the edge of nanostructures. Their amplitude distributions vary with the density of nanostructures as well as distance to the edge. Further, anomalous coherence signals with two fringe envelopes are produced, which result in inaccurate measurement results. We attributed this phenomenon to the complex light modulation by the nanostructures. When referring to the anomalous coherence signals for the top of the nanostructures, one envelope is produced by the top, and the other is produced by the bottom; however, it is difficult to distinguish these two, which is the same case for the bottom of the nanostructures. To automatically solve these obstacles, a signal processing method, which integrates the image segmentation technology to identify and divide the anomalous coherence signals, along with a Morlet wavelet transform to extract the fringe envelope, suitable for any measured area of the dielectric metasurface, is proposed. One metasurface belt consisting of seven kinds of nanopillars with varying arrayed densities that produce different coherence signals is measured. The diameter distribution ranges from 500 to 1250 nm with a constant height of 1850 nm. The local periods in the X and Y directions are 3020 and 1740 nm, respectively. Measurement results demonstrate the validity of the proposed method for spurious fringes processing.

Automated fast computational adaptive optics for optical coherence tomography based on a stochastic parallel gradient descent algorithm.

The transverse resolution of optical coherence tomography is decreased by aberrations introduced from optical components and the tested samples. In this paper, an automated fast computational aberration correction method based on a stochastic parallel gradient descent (SPGD) algorithm is proposed for aberration-corrected imaging without adopting extra adaptive optics hardware components. A virtual phase filter constructed through combination of Zernike polynomials is adopted to eliminate the wavefront aberration, and their coefficients are stochastically estimated in parallel through the optimization of the image metrics. The feasibility of the proposed method is validated by a simulated resolution target image, in which the introduced aberration wavefront is estimated accurately and with fast convergence. The computation time for the aberration correction of a 512 × 512 pixel image from 7 terms to 12 terms requires little change, from 2.13 s to 2.35 s. The proposed method is then applied for samples with different scattering properties including a particle-based phantom, ex-vivo rabbit adipose tissue, and in-vivo human retina photoreceptors, respectively. Results indicate that diffraction-limited optical performance is recovered, and the maximum intensity increased nearly 3-fold for out-of-focus plane in particle-based tissue phantom. The SPGD algorithm shows great potential for aberration correction and improved run-time performance compared to our previous Resilient backpropagation (Rprop) algorithm when correcting for complex wavefront distortions. The fast computational aberration correction suggests that after further optimization our method can be integrated for future applications in real-time clinical imaging.

Nonnull interferometric testing of spherical gratings under Littrow conditions with opposite diffraction orders.

Diffracted wavefront measurements are qualitative and comprehensive verifications for the spherical grating that was manufactured to specifications. Direct interferometric testing of the diffracted wavefront is convenient and implemented by tilting the spherical grating at a Littrow angle to obtain autoreflection and then results in a nonnull interferometric testing configuration. The diffracted wavefront of the spherical grating contains not only wavefront errors induced by the manufacturing imperfections but also inherent wavefront contributions from the autoreflection testing setup. The magnitudes of the latter are affected by both the spherical substrate and the groove pattern. Through the analysis of geometric aberrations of spherical gratings, the groove pattern contributions are demonstrated to be contrary for the opposite diffraction orders. A nonnull interferometric testing of spherical gratings is proposed without foreknowledge of the groove pattern, in which the wavefront errors contributed only by the manufacturing imperfections are derived from dual measurements under Littrow conditions with opposite diffraction orders. Simulations are implemented for varied line spacing (VLS) spherical gratings with an F-number slower than 1.5 and groove density varying from 150 to 300 lp/mm, and the residual error less than 0.004λ RMS is obtained. The residual misalignment error after conventionally removing defocus and tilt is further analyzed and discussed. A VLS grating in which the NA is 0.13 and groove density is 200 lp/mm is chosen as an experimental sample, and the diffracted wavefront error with 0.018λ RMS is obtained.

Subaperture stitching interferometry based on the combination of the phase correlation and iterative gradient methods.

Subaperture stitching interferometry (SAS) is an important method for map testing of large aperture optical components, in which a mechanical structure is often employed for the testing of each subaperture. By eliminating the phase deviation of the corresponding points in the overlapping regions of every adjacent subaperture, the whole aperture map can be obtained. Accurate subaperture positioning is an important guarantee for precise stitching. In this paper, a hybrid optimization algorithm is proposed to realize subpixel-level positioning accuracy in SAS based on the combination of the phase correlation and iterative gradient methods. The phase correlation method is adopted to calculate the pixel-level positioning deviation first, and the subpixel deviation is derived and then corrected by iterative optimization through the gradient method. The subpixel-level positioning accuracy of the proposed optimization algorithm is verified by simulations and a 76.2 mm off-axis parabolic mirror is chosen as an experimental testing sample. The surface map obtained from the proposed hybrid optimization method is consistent with the full aperture testing result, which also verifies that the proposed optimization algorithm is a powerful tool with subpixel-level positioning accuracy in SAS testing.

Construction method through multiple off-axis parabolic surfaces expansion and mixing to design an easy-aligned freeform spectrometer.

The classic Czerny-Turner spectrometer consists of a plane grating and two spherical mirrors. The optical path geometry adopted for incident and grating dispersed light is off-axis reflection, so the spherical collimating and focusing mirrors introduce coma and astigmatism. The conventional configuration is asymmetrical for coma automatic compensation, but suffers from astigmatism. We substitute the off-axis parabolic (OAP) surfaces for spherical surfaces of the collimating mirror and each sub-region of the focusing mirror, to achieve an aberration free configuration. The multiple OAP surfaces are then expanded and mixed, to construct a freeform surface integrating the collimating and focusing mirrors into a single element. Results show that a 0.1 nm spectral resolution is achieved over a bandwidth of 400 nm centered at 800 nm, in the designed spectrometer comprised of a plane grating and one freeform mirror. The construction method is advantageous to integrated optic design, and the resulting freeform mirror spectrometer is compact, and simplifies manufacture and alignment.

Wavefront propagation based on the ray transfer matrix and numerical orthogonal Zernike gradient polynomials.

The aberrated wavefront propagates along its normal. Both the magnitude and boundary change after the propagation. Wavefronts characterized by Zernike coefficients and a normalized pupil radius can also be represented by a bundle of feature rays normal to the local surface. A ray transfer matrix parameterized by the pupil radius and propagation distance is proposed to transfer these feature rays to obtain the slope and position data of the propagated feature rays. Numerical orthogonal Zernike gradient polynomials are derived to reconstruct the wavefront from the discrete data by using a numerical method. Two aberrated wavefronts are performed as examples to validate the accuracy and flexibility of the proposed numerical method.

Customized design and efficient fabrication of two freeform aluminum mirrors by single point diamond turning technique.

The precision single point diamond turning technique has been a promising technology for generating small and medium-sized freeform optical elements with high surface quality. In this paper, we present an extremely off-axis freeform optical system with a large 10.0 mm pupil diameter and a low 3.0 F-number over a wide 28° field of view. It is composed of two freeform aluminum mirrors, which are fabricated efficiently by the single point diamond turning machine. The manufacturing strategy and parameters are estimated rationally and comprehensively, based on the freeform surface characters. The freeform aluminum mirror surface can reach submicron surface accuracy and achieve nanometer surface roughness. The final assembled prototype of the off-axis two-mirror freeform display optical system has the advantages of compactness, a broad spectrum, and good display imaging performance.

Mapping geometry in Fizeau transmission spheres with a small f-number.

The ideal mapping geometry in a Fizeau interferometer is to map equal height increments on a flat object and equal angle increments on a spherical surface to equal heights on the detector. So the initial intent of the optical design of Fizeau transmission spheres (TSs) is to provide R-θ mapping geometry for equal angle increments. The corresponding unequal heights mapping will introduce retrace error as coma when linear carrier fringes exist. On the contrary, equal heights mapping with R-sin θ mapping geometry will avoid linear carrier fringes induced coma error. These two different mapping geometries conflict especially for the TS with a small f-number. In this paper, we will first explore the design and the performance of the f/0.75 TS according to the two different mapping geometries, and then evaluate the mapping geometry for the commercial ZYGO f/0.75 TS, and give some engineering notes for the designers, the metrologists, and the fabricators in the optical laboratory.

Generalized shift-rotation absolute measurement method for high-numerical-aperture spherical surfaces with global optimized wavefront reconstruction algorithm.

In this paper, a generalized shift-rotation absolute measurement method is proposed to measure the absolute surface shape of high-numerical-aperture spherical surfaces. Based on the wavefront difference method, the high order misalignment aberrations can be removed from the measurements. Our generalized shift-rotation absolute measurement process only needs one rotational measurement position and one translational measurement position. A wavefront reconstruction method based on the self-adaptive differential evolution algorithm is proposed to calculate the Zernike polynomials coefficient ai of the absolute surface shape Wtest(x,y), the rotation angle Δθ, the translation δx along the x axis, and the translation δy along the y axis. The translation error and rotation error in other absolute measurement methods are avoided using our generalized shift-rotation absolute measurement method. Experimental absolute results of the test surface and reference surface are given and the difference of reference surface shapes between two testings in experiments is 0.12 nm root mean square.

Large radius of curvature measurement based on virtual quadratic Newton rings phase-shifting moiré-fringes measurement method in a nonnull interferometer.

We have proposed a virtual quadratic Newton rings phase-shifting moiré-fringes measurement method in a nonnull interferometer to measure the large radius of curvature for a spherical surface. In a quadratic polar coordinate system, linear carrier testing Newton rings interferogram and virtual Newton rings interferogram form the moiré fringes. It is possible to retrieve the wavefront difference data between the testing and standard spherical surface from the moiré fringes after low-pass filtering. Based on the wavefront difference data, we deduced a precise formula to calculate the radius of curvature in the quadratic polar coordinate system. We calculated the retrace error in the nonnull interferometer using the multi-configuration model of the nonnull interferometric system in ZEMAX. Our experimental results indicate that the measurement accuracy is better than 0.18% for a spherical mirror with a radius of curvature of 41,400 mm.

Carrier squeezing interferometry with π/4 phase shift: phase extraction in the presence of multi-beam interference.

Multi-beam interference exists in testing high-reflectivity surfaces with a Fizeau interferometer. In this paper, the multi-beam interference intensity was estimated as the sum of the first six order harmonics using the Fourier series expansion. Then, by adopting carrier squeezing interferometry with a π/4 phase shift, an algorithm was proposed to extract the phase from multi-beam interferograms. To ensure the separation of the lobes of phase-shift errors and the phase in the frequency domain, conditions of the necessary linear carrier in the proposed algorithm were derived. Simulation results indicated that the phase retrieving precision is better than PV 0.008λ and RMS 0.001λ, even when the reflection coefficient of the test surface is as high as 0.9 and the phase shift varies within π/4±π/20. Compared with the other algorithms, the proposed algorithm for multi-beam interference was validated by its good performance in the experiments, especially when the phase-shift error exists.

Alignment analyses of a galvanometer-based scanner in free-space Fourier domain optical coherence tomography.

Free-space Fourier domain optical coherence tomography is adopted for biomedical imaging with ultrahigh resolution, in which the setup consists of an interferometer and a spectrometer. Two-dimensional lateral sampling in the sample arm of the interferometer is achieved by using a galvanometer-based scanner. Optical path difference (OPD) drift in the full scan field of view is observed in the assembly process of the scanner. A galvo mirror mount offset with respect to the rotation axis is demonstrated as the derivation of this OPD drift by both geometric analyses and model building. Then, an iterative assembly process of the scanner is proposed with the OPD drift taken as the alignment criteria.

Modal wavefront estimation from its slopes by numerical orthogonal transformation method over general shaped aperture.

Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.

Applying slope constrained Qbfs aspheres for asphericity redistribution in the design of infrared transmission spheres.

An achromatic transmission sphere (TS) must be provided as an accessory for an infrared interferometer with a broadband wavelength channel. When designing an F/0.75 infrared TS using the one-asphere model of the two-lens-only package, the first surface was optimized as a strong asphere beyond our current testing ability, using either a slope-unconstrained Qbfs asphere equal to a conventional even asphere, or using a slope-constrained Qbfs asphere. We applied the asphere-splitting theory to redistribute the asphericity. Adopting only the slope-constrained Qbfs asphere, the strong asphere was successfully split into two weaker aspheres within our testing ability, getting a more manufacturable two-asphere TS solution. Besides, the TS solution with two weaker aspheres presented the same sensitivity to assembly errors compared to the solution with one strong asphere, as demonstrated by the analysis of the wavefront aberrations of the TS solutions. This phenomenon coincides well with the asphere-splitting theory, given that the total asphericity remains almost equivalent between the two solutions.

Comparative assessment of orthogonal polynomials for wavefront reconstruction over the square aperture.

Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials, 2D Legendre polynomials, Zernike square polynomials and Numerical polynomials. They are all orthogonal over the full unit square domain. 2D Chebyshev polynomials are defined by the product of Chebyshev polynomials in x and y variables, as are 2D Legendre polynomials. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness. Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.

Application of 3-dimensional printing technology to construct an eye model for fundus viewing study.

OBJECTIVE: To construct a life-sized eye model using the three-dimensional (3D) printing technology for fundus viewing study of the viewing system. METHODS: We devised our schematic model eye based on Navarro's eye and redesigned some parameters because of the change of the corneal material and the implantation of intraocular lenses (IOLs). Optical performance of our schematic model eye was compared with Navarro's schematic eye and other two reported physical model eyes using the ZEMAX optical design software. With computer aided design (CAD) software, we designed the 3D digital model of the main structure of the physical model eye, which was used for three-dimensional (3D) printing. Together with the main printed structure, polymethyl methacrylate(PMMA) aspherical cornea, variable iris, and IOLs were assembled to a physical eye model. Angle scale bars were glued from posterior to periphery of the retina. Then we fabricated other three physical models with different states of ammetropia. Optical parameters of these physical eye models were measured to verify the 3D printing accuracy. RESULTS: In on-axis calculations, our schematic model eye possessed similar size of spot diagram compared with Navarro's and Bakaraju's model eye, much smaller than Arianpour's model eye. Moreover, the spherical aberration of our schematic eye was much less than other three model eyes. While in off- axis simulation, it possessed a bit higher coma and similar astigmatism, field curvature and distortion. The MTF curves showed that all the model eyes diminished in resolution with increasing field of view, and the diminished tendency of resolution of our physical eye model was similar to the Navarro's eye. The measured parameters of our eye models with different status of ametropia were in line with the theoretical value. CONCLUSIONS: The schematic eye model we designed can well simulate the optical performance of the human eye, and the fabricated physical one can be used as a tool in fundus range viewing research.

Focal length measurement based on the wavefront difference method by a Fizeau interferometer.

A method for measuring the focal length of the lens by a Fizeau interferometer is proposed. Based on the Gaussian imaging equation and the longitudinal displacements of the object point and image point, a precise formula for focal length calculation is deduced. The longitudinal displacement of the object points is determined by the wavefront difference method with a subnanometer resolution. An experimental system for focal length measurements is set up to verify the principle. The sources of uncertainty in measurement are discussed. Both the positive and negative lens experimental results indicate that the measurement accuracy is less than 0.16% under normal experimental environment.

An absolute test for axicon surfaces.

We present a method for absolute testing of axicon surfaces in a null test setup. The absolute test exploits the symmetry of axicons, which allows us to introduce a shift of the surface under test in both the axial and rotational directions while still maintaining the null test condition. With two shifts of the surface under test and four measurements, the interferometer and null optics error can be removed. The absolute surface local deviation can be obtained by wavefront reconstruction with a double-side spiral-path direct integration method. A simulation of the method, including typical systematic as well as statistical errors as input, is presented to estimate the error propagation. Experimental absolute test results of a 90° axicon surface are given.

A generic eye model by reverse building based on Chinese population.

The human eye has ethnic difference, the existing typical eye models are based on western eyes. A generic eye model based on Chinese population is presented for the first time. The statistical analyzed ocular parameters based on measured data are used for the initial generic eye model, and the wavefront aberration data obtained at two different pupil diameters are used for reproduction by optimizing the initial generic eye model. The differences and similarities between Chinese generic eye model and western eye models are given. The Chinese generic eye model provides a suitable model for the related further researches and applications on Chinese eye.