Phase errors in high line density CGH used for aspheric testing: beyond scalar approximation

Fabrication of a large computer-generated hologram with high diffraction efficiency and high accuracy by scanning homogenization etching

The diffraction efficiency, defined as the ratio of diffracted power to incident power, is one of the key working indicators for a computer-generated hologram (CGH). The CGH with high diffraction efficiency could suppress stray light and eliminate ghost images, thus improving interferometric performance in aspherical testing of low-reflectivity or large off-axis distance surfaces. However, the high-efficiency CGH is hard to precisely fabricate by traditional reactive ion etching and focusing ion beam, because it requires high etching depth with a high uniformity and sub-nanometric roughness in the glass, especially in the fabrication of a large CGH with an aperture of up to 300 mm. In this study, fabrication of the above-mentioned CGH was demonstrated via what we believe to be a new method called scanning homogenization etching (SHE), in which the ion source with a Gaussian energy distribution accurately scans the glass surface to realize homogenization etching. Different from controlling dwell time at each etching point, this paper proposes to control the scanning rate to achieve not only uniform but also quantitative depth removal in a single scan. Moreover, the depth errors in deep etching across the whole glass surface can be remarkably reduced due to homogenization effects introduced by multiple scanning etching. Finally, the target etching depth of 692.3 nm with an etching uniformity of 2.2% in the etching of a 300 mm CGH was achieved. The roughness of the etched and unetched area both have Ra values of 0.3 nm. The diffraction efficiency of working order is 39.998%, achieving 98.6% of the theoretical diffraction efficiency. In addition, the SHE is not limited by the aperture of the ion source, so it can achieve even larger diffractive optical elements with high diffraction efficiency and high accuracy.

Aspheric surface measurement by absolute wavelength scanning interferometry with model-based retrace error correction

This paper presents a non-nulling absolute interferometric method for fast and full-area measurement of aspheric surfaces without the necessity of any mechanical movement. Several single frequency laser diodes with some degree of laser tunability are used to achieve an absolute interferometric measurement. The virtual interconnection of three different wavelengths makes it possible to accurately measure the geometrical path difference between the measured aspheric surface and the reference Fizeau surface independently for each pixel of the camera sensor. It is thus possible to measure even in undersampled areas of the high fringe density interferogram. After measuring the geometrical path difference, the retrace error associated with the non-nulling mode of the interferometer is compensated for using a calibrated numerical model (numerical twin) of the interferometer. A height map representing the normal deviation of the aspheric surface from its nominal shape is obtained. The principle of absolute interferometric measurement and numerical error compensation are described in this paper. The method was experimentally verified by measuring an aspheric surface with a measurement uncertainty of λ/20, and the results were in good agreement with the results of a single-point scanning interferometer.

Solving optimal carrier frequencies of a CGH null compensator through a double-constrained searching method based on iterative ray-tracings

Interferometry based on a computer-generated hologram (CGH) null compensator is a general method for high-precision metrology of aspherics. Because the most commonly used CGHs are the Ronchi type with only two quantization steps, tilt and defocus carrier frequencies must always be introduced to separate the disturbing diffraction orders (DDOs). Determining the amount of carrier frequencies is a pivotal but difficult issue in the CGH design process. Previous studies have only drawn qualitative conclusions or obtained some approximate results under specific conditions. This paper proposes a double-constrained searching method based on iterative ray-tracings, which can directly and accurately give the optimal combination of tilt and defocus carrier frequencies, as long as the aspheric under test is a concave one and has an analytical expression. The optimal carrier frequencies solved by the proposed method will minimize the line density of the CGH on the premise of separating all DDOs, which will reduce the cost and difficulty of fabrication as much as possible. The proposed method is almost error-free and holds a clear advantage over the previous methods in terms of versatility. Several typical design examples are presented to verify the feasibility and versatility of the proposed method. Its accuracy is also verified through making comparisons of the ray-tracing results between another method and Zemax models based on these examples.

Measurement of rotationally symmetric aspherical surfaces with the annular subaperture stitching method

The annular subaperture stitching method is an effective method for testing rotationally symmetric aspherical surfaces. To create a full aperture map of this kind of asphere, we propose an annular subaperture stitching algorithm. The algorithm is based on triangulation interpolation theory, least-square fitting method, and ray tracing method. We first show the principle of our stitching algorithm. Then, the performance of the proposed method is analyzed by simulation testing to evaluate the accuracy of the above algorithm. In the end, the experiment is given to demonstrate the correctness of our method. Both the simulation and experiment results show that the introduced method is quite effective for the testing of rotationally symmetric aspherical surfaces.

Fabrication of a high-accuracy phase-type computer-generated hologram by physical vapor deposition

With the development of optical systems used in astronomical and earth observation, aspherical and free-form surfaces are increasingly used because they are lightweight and have improved image quality. As a highly accurate, aberrationless technique, computer-generated hologram (CGH) plays an important role in wavefront testing. At present, the main way to fabricate phase CGH is reactive ion etching, which suffers from low accuracy. To improve the accuracy, physical vapor deposition (PVD) is applied in the fabrication of phase CGH. The wavefront errors of PVD-fabricated phase CGH were analyzed. Testing results indicate that the wavefront error of the CGH is 0.020λ root mean square (RMS), mainly caused by the machine tool orthogonality error rather than the PVD process. The diffraction efficiency of the +1st order is 22.4%.

Analysis of spurious diffraction orders of computer-generated hologram in symmetric aspheric metrology

Computer-generated hologram (CGH) has been widely used as a wavefront compensator in symmetric aspheric metrology. As a diffractive element, it generates different diffraction orders, but only the 1st-order diffraction is used to test aspheric surface. The light from spurious diffraction orders (SDO) will produce many high-frequency fringes in interferogram and reduce measurement accuracy. In this paper, we regard the CGH null system as an imaging system and develop an aberration model in Seidel formalism to analyze the SDO. This model has the advantage to address the difference between the SDO (k1, k2) and (k2, k1). We consider the effect of the pupil distortion so that our model can analyze the SDO with a large pupil distortion. We derive the condition to ensure the 2^nd-order and 4^th-order aberrations have the same sign and calculate the minimum defocused distance (power carrier frequency) of CGH. According to the marginal-ray heights (h₁andh₃) on the CGH in the first and second passes, we determine the condition that the SDO covers the whole CGH in the second pass. We analyze the SDO of 4 CGH designs and compare the results from our aberration model with these from real ray trace. These results validate that our aberration model is feasible whether the aspheric part is convex or concave and whether CGH is inside or outside the focus of the transmission sphere.

Optimal design of tilt carrier frequency computer-generated holograms to measure aspherics

Computer-generated holograms (CGHs) provide an approach to high-precision metrology of aspherics. A CGH is designed under the trade-off among size, mapping distortion, and line spacing. This paper describes an optimal design method based on the parametric model for tilt carrier frequency CGHs placed outside the interferometer focus points. Under the condition of retaining an admissible size and a tolerable mapping distortion, the optimal design method has two advantages: (1) separating the parasitic diffraction orders to improve the contrast of the interferograms and (2) achieving the largest line spacing to minimize sensitivity to fabrication errors. This optimal design method is applicable to common concave aspherical surfaces and illustrated with CGH design examples.

Limits of diffractometric reconstruction of line gratings when using scalar diffraction theory

We study errors that occur in geometry and phase reconstruction when using scalar diffraction theory in line gratings with periods below 10 μm. The application of those gratings in so-called computer-generated holograms in high-precision interferometric testing of aspheres and free-form surfaces imposes high demands on the generated phase, leading to error budgets in the range of λ/100. Using rigorous simulations as references, we identify the limits where scalar diffraction theory fails to accurately describe grating geometries and identify the significant error mechanisms.