Null Optics Research Articles

Research on Nanometer Precision Measurement Method of High Order Even Aspheres

Optical aspheres are demanded with extremely high precision to meet functional requirements in space telescopes, extreme ultraviolet lithography, and other modern large optical systems. The nano-precision fabrication of optical aspheres requires high-precision measurements to guide deterministic optical processing. Null test is the preferred method for high-precision measurements. Null optics are required to compensate for the incident wavefront in the null test of optical aspheres. However, wavefront aberrations caused by the transmission flat or transmission sphere of interferometer and null optics can limit measurement accuracy and need to be separated. A nano-precision measurement method is proposed for the even optical aspheres of high order in this paper. A computer-generated hologram is used as a null optic to realize a null test on optical aspheres. Mapping distortion correction is performed on the measurement results to ensure that the transverse coordinates of the measurement results correspond correctly to those of the test surface. Absolute testing is applied to separate the wavefront aberrations caused by the computer-generated hologram and interferometer optics. Finally, the results obtained by this method were used to guide deterministic optical processing, enabling the nano-precision fabrication of optical aspheres.

Positioning of the test surface in a CGH null test by cat's eye interference

Interferometric null test of optical aspheres and freeforms is one of the most established methods, in spite of recognized limitations related to positioning of the test surface. Slight misalignment of the surface in the null test system can introduce remarkable wave aberrations. Current approaches for positioning the test surface are based on a series of fiducial marks or retroreflectors set at given spots around the test surface, which consequently suffer the problem of datum transformation. We propose a method for positioning the test surface in a null test by cat's eye interference without any fiducial marks or retroreflectors. A computer-generated hologram (CGH) is used as null optics fabricated with null test pattern, alignment pattern and positioning pattern. A part of test beam with relatively small f/number is diffracted through the positioning pattern and then focuses on certain spots of the test surface. The cat's eye reflection from the test surface returns to the interferometer and interferes with the reference beam. It is then possible to precisely position the test surface by measuring the wavefront error of the cat's eye interference. The design method for such a CGH positioning pattern is presented, following the physical constraint that the surface normal at the cay's eye reflection is right the angular bisector of the positioning beam. Analysis on the positioning performance is then presented to show the sensitivity to misalignment including defocus, lateral shift and tip-tilt, which at last is experimentally verified by measuring an even asphere with a CGH.

Absolute testing of rotationally symmetric surfaces with computer-generated holograms

Extremely high accuracy is demanded for optics working at very short wavelength. Interferometric testing of optical aspheres or freeform surfaces requires null optics, typically computer-generated holograms (CGHs), to balance the wave aberrations. The measurement uncertainty is primarily limited by the accuracy of the test wavefront, which is predominantly influenced by the CGH and the interferometer optics. Absolute testing is fundamental to achieving accuracy much higher than that of the test wavefront through error separation. This paper presents a method for absolute testing of rotationally symmetric surfaces with CGH null optics. The basic assumption is that the off-axis hologram fabricated by raster scanning beam writing has negligible error of rotationally symmetric component due to pattern error of the CGH. Consequently, the wavefront error contributed by the CGH and the transmission flat can be completely separated from the absolute surface shape by combining the N-position method and the shift-rotation method. A theoretical model for absolute testing is proposed under the assumption. Experimental cross test is then presented to validate the method with sub-nanometer uncertainty. The assumption is further confirmed by characterizing the fabrication error of the hologram structures using a white light interferometer. Finally, the effect of noise, translation error, rotation error and eccentricity of rotation on the absolute testing is analyzed.

Flexible and high-resolution surface metrology based on stitching interference microscopy

Small optics of 10 mm-scale are widely used in miniature cameras with increasingly sophisticated surface shape. The surface features ranging from low to high frequency need to be measured efficiently for in-process quality control. Wavefront interferometry is applicable with specifically designed null optics for each surface, but not preferred for small optics because the surface radius is approaching the shortest measurable radius limited by the interferometer optics. We present a strategy for flexible surface metrology based on stitching interference microscopy. The full aperture surface is divided into a series of small subapertures with reduced size and slope which are then measurable with a standard interference microscope. The measurement principle is introduced and a method for subaperture design is proposed by checking the subaperture size and slope after coordinate transformation. A simultaneous stitching algorithm is proposed to minimize the overlapping deviations with overlapping points and subaperture configurations determined automatically in a way of alternating optimization. Instrumentation is easy by integrating a multi-axis platform with a commercial interference microscope. Both the lateral and the vertical scales need to be calibrated to give accurate subaperture measurement of three coordinates. The capability of the instrument is finally verified through a series of experiments compared with cross tests such as the null test and the stylus profilometry. It shows high flexibility to measure a variety of complex surfaces including the high-order even asphere, the “gull-wing” surface and microstructured curved surfaces without using any auxiliary null optics. The instrument enables “one-stop” measurement of surface form, waviness and roughness simultaneously.

Fiducial free correction of mapping distortion in null test of aspheres and freeforms

Aspheres and freeform surfaces are well measured with null optics but the transverse coordinates on the test surface are not linear with the pixels on the imaging plane. Such a mapping distortion must be corrected before the measured surface error map is used for corrective figuring process. Conventional mapping distortion correction involves tediously finding the coordinates of a sequence of fiducial marks on both the test surface and the imaging plane. This paper presents a pure ray tracing method without use of fiducial marks by introducing a dummy spherical surface between the point source and the null optics, acting like a virtual reference sphere (RS). It is linearly mapped to the imaging plane with the linear scale easily calibrated. On the other hand, the virtual RS is exactly mapped to the test surface by tracing a bundle of dense rays. This virtual surface thus bridges the gap between coordinates on the imaging plane and those on the test surface. Two experimental examples of null test of on-axis and off-axis aspheres demonstrate the effectiveness of mapping distortion correction.

Development of a concave freeform surface measurement using transverse translation-diverse phase retrieval

Transverse translation-diverse phase retrieval (TTDPR), a ptychographic wavefront-sensing technique, is a viable method for freeform optical surface metrology due to its relatively simple hardware requirements, flexibility, and demonstrated accuracy in other fields. In TTDPR, a subaperture illumination pattern is scanned across an optic under test, and the reflected intensity is gathered on an array detector near focus. A nonlinear optimization algorithm is used to reconstruct the wavefront aberration at the test surface from which we can solve for surface error, using intensity patterns from multiple scan positions. TTDPR is an advantageous method for aspheric and freeform metrology because measurements can be performed without null optics. We report on the development of a concave freeform mirror measurement using this technique. Simulations were performed to test algorithmic performance as a function of various parameters, including detector signal-to-noise ratio and position uncertainty of the illumination, with <λ / 100 root-mean-square (rms) wavefront-sensing error achieved in most cases (λ = 632.8 nm). An experimental measurement is then demonstrated, and results of reconstructed surface form and midspatial frequency error are presented. Surface reconstructions from two disjoint datasets agree to 13-nm rms, or λ / 50 at λ = 632.8 nm.

Null test of large convex aspheres by subaperture stitching with replaceable holograms

Null test is preferred for measuring surface error of aspheres. However for large convex aspheres, it is challenging to fabricate larger null optics. Subaperture stitching is hence introduced but requires different null optics for subapertures of different off-axis distances. It gets easier by using replaceable computer generated holograms (CGHs). In this paper, a convex high-order even asphere is taken as an example and multiple CGHs are designed for null test of different subapertures. The same transmission flat and tilt carrier are used to facilitate alignment of CGHs. Disturbance orders of diffraction are confirmed to be separated by the tilt carrier without ghost interference. A configuration-based stitching algorithm is applied to successfully retrieve the full aperture surface error. The CGH substrate error is calibrated out before stitching optimization by measuring its transmitted wavefront with a high precision reference flat. Finally, the stitching result is verified by through-the-back null test of the same asphere.

Annular subaperture interferometry for high-departure aspheres using paraboloidal parameterization.

A low-cost technique is presented for constructing stitched Fizeau interferometric measurements of high-spherical-departure concave aspheres of arbitrary conic constant without the use of null optics. The optical test configuration assembles the surface figure of the asphere using subapertures parameterized as variances from best-fit paraboloids. Subtracting the optical path difference between each idealized paraboloid and the corresponding annulus of measured data removes the annular wavefront aberrations without the need to fit Zernike polynomials. The proof-of-concept measurement and reconstruction of a 250mm diameter diamond-turned ellipsoidal mirror with more than 3000 waves of spherical departure are reported. The presented technique is an inexpensive addition to the array of tools used to measure large-aperture aspheres and high-departure freeform optics.

Adaptive wavefront interferometry for unknown free-form surfaces

The primary problem of conventional wavefront interferometers is limited dynamic range. Unknown free-form surface figure error with large amplitude or slope is not measurable for too dense or invisible fringes. To troubleshoot this problem, we propose adaptive wavefront interferometry (AWI). AWI utilizes a wavefront sensor-less adaptive optics (AO) subsystem to intelligently speculate and compensate the unknown free-form surface figure error. In this subsystem, adaptive null optics is utilized to iteratively generate adaptive wavefronts to compensate the unknown severe surface figure error. The adaptive null optics is close-loop controlled (i.e., wavefront sensor-less optimization algorithms are utilized to control it by real time monitoring the compensation effects to guarantee convergence of the iteration). Ultimately, invisible fringes turn into resolvable ones, and null test is further realized. To demonstrate the feasibility of AWI, we designed one spatial light modulator (SLM) based AWI modality as an example. The system is based on a commercial interferometer and is easy to establish. No other elements are required besides the SLM. Principle, simulation, and experiments for the SLM based AWI are demonstrated.

Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror

Partial null interferometry without using any null optics is proposed to measure a concave freeform Zernike mirror. Oblique incidence on the freeform mirror is used to compensate for astigmatism as the main component in its figure, and to constrain the divergence of the test beam as well. The phase demodulated from the partial nulled interferograms is divided into low-frequency phase and high-frequency phase by Zernike polynomial fitting. The low-frequency surface figure error of the freeform mirror represented by the coefficients of Zernike polynomials is reconstructed from the low-frequency phase, applying the reverse optimization reconstruction technology in the accurate model of the interferometric system. The high-frequency surface figure error of the freeform mirror is retrieved from the high-frequency phase adopting back propagating technology, according to the updated model in which the low-frequency surface figure error has been superimposed on the sag of the freeform mirror. Simulations verified that this method is capable of testing a wide variety of astigmatism-dominated freeform mirrors due to the high dynamic range. The experimental result using our proposed method for a concave freeform Zernike mirror is consistent with the null test result employing the computer-generated hologram.

Mode conversion efficiency to Laguerre-Gaussian OAM modes using spiral phase optics.

An analytical model for the conversion efficiency from a TEM00 mode to an arbitrary Laguerre-Gaussian (LG) mode with null radial index spiral phase optics is presented. We extend this model to include the effects of stepped spiral phase optics, spiral phase optics of non-integer topological charge, and the reduction in conversion efficiency due to broad laser bandwidth. We find that through optimization, an optimal beam waist ratio of the input and output modes exists and is dependent upon the output azimuthal mode number.

Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors

Interferometric test of large convex aspheres with high accuracy is still an urgent problem. Adopting a large-aperture null corrector is usually inevitable, and it is imperative to certify the large-aperture null optics in case incorrect final shape of the aspheric mirror is obtained. Moreover, it is necessary to calibrate and remove errors of the large-aperture null lens and trace errors to surfaces that are easy to obtain high accuracy. For these purpose, this paper presents an in situ calibrated null test method. The large-aperture null corrector for convex aspheric mirror is verified and calibrated by a wisely designed small aperture certifying null whose surfaces are either flat or spherical which are easy to be fabricated, measured and assembled. A redundant test is implemented by a Zygo VeriFire Asphere interferometer for cross test. Compared with existing methods of certificating null correctors which utilize an expensive CGH or a self-aligning aspherical mirror, the presented method finally traces errors of a large-aperture null corrector to small aperture flat and spherical surfaces thus costs are expected to be saved dramatically.

Absolute test for cylindrical surfaces using the conjugate differential method

An absolute testing method for cylindrical surfaces is presented in a null test setup with a computer-generated hologram. The absolute test exploits the symmetry of cylinders, which allows us to introduce a certain shift of the test surface both parallel to and rotated about the centerline while the null test condition is still maintained. With two shifts of the cylindrical surface, four measurements belonging to two groups in conjugate positions can be accomplished to obtain the absolute differential map with the interferometer and null optics errors removed. The absolute surface can be obtained by wavefront reconstruction from local differential data. A simulation of the method is presented to estimate the error propagation. Experimental absolute test results of a concave cylindrical surface with 100-mm radius are given. The measured profiles are compared with those obtained from a commercial profiler, showing a difference of less than 15 nm (root-mean-square).

High-accuracy aspheric x-ray mirror metrology using Software Configurable Optical Test System/deflectometry

The Software Configurable Optical Test System (SCOTS) uses deflectometry to measure surface slopes of general optical shapes without the need for additional null optics. Careful alignment of test geometry and calibration of inherent system error improve the accuracy of SCOTS to a level where it competes with interferometry. We report a SCOTS surface measurement of an off-axis superpolished elliptical x-ray mirror that achieves <1 nm root-mean-square accuracy for the surface measurement with low-order term included.

Measurement of mild asphere by digital plane

In order to test mild aspheric surface directly without other null optics, the digital plane method is proposed. When departure of the tested aspheric surface is mild, a sphere mirror can be used as the reference surface. The phase distribution can be measured swiftly by the digital interferometer. The surface error can be obtained by subtracting the theory wavefront error (the value of the digital plane) from the phase data and eliminating the translation errors through least-squares fitting. The basic principle and theory of this method are analyzed and researched, and the testing model and flow chart are established. Meanwhile the experiment is carried out with a mild aspheric mirror by this method, the PV and RMS of the surface error are 0.173 and 0.018 ( is 632.8 nm), respectively. We also design and make a null corrector to the asphere for contrast, the differences in PV and RMS error between them are 0.026 and 0.001, respectively.

Synthetic phase-shifting for optical testing: Point-diffraction interferometry without null optics or phase shifters

An innovative iterative search method called the synthetic phase-shifting (SPS) algorithm is proposed. This search algorithm is used for maximum-likelihood (ML) estimation of a wavefront that is described by a finite set of Zernike Fringe polynomials. In this paper, we estimate the coefficient, or parameter, values of the wavefront using a single interferogram obtained from a point-diffraction interferometer (PDI). In order to find the estimates, we first calculate the squared-difference between the measured and simulated interferograms. Under certain assumptions, this squared-difference image can be treated as an interferogram showing the phase difference between the true wavefront deviation and simulated wavefront deviation. The wavefront deviation is the difference between the reference and the test wavefronts. We calculate the phase difference using a traditional phase-shifting technique without physical phase-shifters. We present a detailed forward model for the PDI interferogram, including the effect of the finite size of a detector pixel. The algorithm was validated with computational studies and its performance and constraints are discussed. A prototype PDI was built and the algorithm was also experimentally validated. A large wavefront deviation was successfully estimated without using null optics or physical phase-shifters. The experimental result shows that the proposed algorithm has great potential to provide an accurate tool for non-null testing.

Aspheric surface measurement based on sub-aperture stitching interferometry

In order to test convex aspheric surfaces without the aid of other null optics, a novel method combined sub-aperture stitching and interferometry called SSI (sub-aperture stitching interferometry) is introduced. In this letter, the theory, basic principle, and flow chart of SSI are researched. A synthetical optimization stitching mode and an effective stitching algorithm are established based on homogeneous coordinate's transformation and simultaneous least-squares fitting. The software of SSI is devised, and the prototype for testing of large aspheres by SSI is designed and developed. The experiment is carried out with five subapertures for a convex silicon carbide (SIC) aspheric mirror with a clear aperture of 130 mm. The peak-to-valley (PV) and root-mean-square (RMS) error are 0.186 \lambda and 0.019 \lambda, respectively. For the comparison and validation, the TMA system which contained the convex asphere is tested by interferometry. The wavefront error of the central field of the optical system is 0.068 \lambda RMS which approaches to diffraction limitation. The results conclude that this technique is feasible and accurate. It enables the non-null testing of aspheric surfaces especially for convex aspheres.

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.

Binary amplitude holograms made from dyed photoresist

A method for fabricating binary amplitude holograms from a dyed photoresist is described. It is of particular interest for holograms that are used as null optics in the form metrology of aspheric surfaces and wavefronts. A pigment that strongly absorbs light near 633 nm was dissolved in a positive photoresist and the dyed resist was spun onto silica glass substrates. Stable resist layers were obtained that were essentially opaque at 633 nm with little effect on the transmittance of the resist in the UV. A Fresnel zone plate was fabricated from the dyed resist layer using contact lithography, and its performance was demonstrated at 633 nm.

Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave

A simple interferometric method is presented for determining the parent radius of curvature (ROC) and conic constant (CC) of a conic surface. This method compares the test surface with a spherical reference wavefront having a ROC equal to the local sagittal, medial, or tangential ROC. The measured wavefront aberrations, particularly the astigmatism and coma, and local radii are used to determine the parent ROC and CC. This method does not require null optics or knowledge of the surface coordinate where the measurement is made.