Due to its potential for pm-level displacement measurement accuracy, orbital angular momentum (OAM) interferometer recently has attracted extensive research interest. However, the displacement range of common OAM interferometer always is limited to submicrometer level due to the phase ambiguity in demodulation. To overcome this limitation, we firstly proposed a polarization-separated dual-wavelength OAM interferometer. The measurement range can be increased by three orders of magnitude through dual-wavelength interferometry. Benefiting from the unique optical structure design of polarization-separated common optical path, dual-wavelength interferograms can be synchronously and stably acquired using a polarization camera. Moreover, the impact of environmental disturbances can be further suppressed by using OAM beams with opposite topological charge. Experimental results show that a 7500.0 nm displacement is measured as 7447.1 nm (0.7% error), while step measurements achieve linearity >0.999. The proposed interferometer can realize a single-frame wide-range displacement measurement, maintaining good stability and measurement accuracy, which may provide a promising technical solution for wide-range high-precision measurement.

In phase-shifting interferometry using narrow-bandwidth light sources, ripple noise introduced by multiple error sources is one of the key factors limiting its wavefront calculation accuracy, this study proposed a generalized ripple suppression algorithm (GRSA) in its standard measurement scenarios. The distribution law of ripple errors caused by different error sources were investigated, and a mathematical model was established to decompose the wavefront shape components. The ripple components in wavefronts can be calculated and directly subtracted via two least-squares fittings on multiple wavefronts containing different ripple errors. Feasibility of the GRSA was verified through simulations and experiments, and the root mean square error of the simulated wavefront reaches 2.1 x 10-5 lambda. Factors affecting the accuracy of phase calculation and several special cases were analyzed, and optimization schemes for algorithm parameters were provided based on the analysis results. The experimental and simulation results show that the GRSA can effectively improve measurement accuracy and repeatability by suppressing the ripple error in the wavefront.

Random sphere absolute measurement is a simple and efficient method. It assumes that the mean of the surface in different regions of the test sphere necessarily converge to the ideal plane. Although a large number of experiments have verified this assumption, the relevant theory has not yet been perfected. In this study, we develop a mathematical model for the random sphere absolute measurement from the perspective of moments of random variables. It shows that the random range of the test spheres significantly affects the repeatability of measurement. Meanwhile, based on this mathematical model, we proposed the multiple sphere random theory. In the experiment, the RMS repeatability for single sphere random absolute measurements is 0.00045 lambda under the condition of large random range and 0.00017 lambda under the condition of small random range. The experiment verifies the influence of the random range on the repeatability of the measurement. Meanwhile, a multiple spheres random absolute measurement experiment was completed, which achieved a similar repeatability to the single sphere random under the condition of poorer surface quality of the test sphere and fewer averaging times. The RMS repeatability is 0.00022 lambda. This method greatly reduced the cost of the sphere absolute measurement.

An iterative algorithm for tilt phase-shift interferograms with inhomogeneous background is proposed in this paper. The regional integration results of the interferograms are fitted with a cosine curve in the time domain to obtain the iterative initial values, then the stepwise iterative optimization is carried out for wavefront phase, background, x-direction tilt shift, and y-direction tilt shift by the least-squares method, respectively, and the phase distribution can be obtained after several rounds of iterations. Feasibility of the algorithm is verified through simulations and experiments. Factors affecting the accuracy of phase calculation are analyzed, and the results show that the proposed method is insensitive to the background light intensity distribution, and there is almost no limit on the number of fringes. The algorithm can accept tilt-shift error with vibration amplitude less than 0.86 pi rad and translational-shift error with vibration amplitude less than 0.43 pi rad.

We propose a new interferometer system, the cascade Twyman interferometer (CTI), that can actively compensate for systematic errors. It adds an extra channel to the Twyman interferometer and uses polarized beam interference to separate the three beams. Subsequently, three interferograms, or two-by-two interferences of the three beams, can be obtained. Finally, the absolute surface distribution of the test mirror is calculated using the three-flat mutual inspection. The simulation showed that the CTI was only affected by the wavefront transmission error, which was negligible. In the experiment, the RMS repeatability of CTI was <10-4 lambda. Meanwhile, compared with the even and odd function methods, the results were similar, with a difference between the Zernike coefficients of <0.0028 lambda. This indicates that CTI is an efficient and accurate interferometer.

A statistical model was constructed to study the effect of random noise in interferometry on the absolute measurement of shift rotation. The results indicated sensitivity to random noise, particularly under low signal-to-noise ratio conditions. Therefore, we propose a low-signal-to-noise-ratio surface reconstruction (LSSR) algorithm, which is a compelling iteration for achieving the surface reconstruction of absolute measurement of shift rotation. In simulations, the peak-to-valley (PV) of the residuals of LSSR reached λ/1000, whereas the classical method could only reach λ/100. In experiments, the PV of the surface distributions reconstructed using LSSR reached λ/40, and the reconstructed surface distributions were consistent regardless of the magnitude of random noise. In addition, compared to the three-flat mutual inspection method, LSSR also exhibited consistency, significantly suppressing the effect of random noise on the absolute measurement of shift rotation.

Objective As the traditional optical and semiconductor industries require higher component accuracy,precision measurement technology has gradually developed in the direction of high accuracy and high stability.As a non-contact high-precision detection method,interferometric test technology has a wide range of applications in the field of measurement.The transmission flat is the core element in interferometers,and its face shape accuracy has a direct influence on testing accuracy.In practice,the mounting datum surface cannot be a standard plane,so both position change and cross-system migration can alter the support conditions of the transmissive flat,which in turn leads to deformation.This reduces the reliability of the calibration results during iterative polishing and subsequent use,seriously affecting the flat's precision and even the interferometer's testing accuracy.To meet the surface shape accuracy requirements of transmission flats in application,a suitable support structure is essential. Methods The flat mounting datum surface can be regarded as a rough surface,while a real rough surface consists of a large number of randomly distributed protruding peaks.Both the movement of the transmissive flat and changes in the mounting datum affect the distribution of force points on the frame.To stabilize the force points on the frame,additional prominent supports are required at fixed positions on the lower surface of the frame,replacing the random protruding peaks of the mounting datum.By combining contact mechanics and the three-point principle for plane definition,we design a three-point support structure for the frame.First,we mount the transmission flat using an appropriate method such as adhesive support.Then,we add three support points under the frame to isolate it from the mounting datum and fix the load points,thus reducing the influence of the datum surface.Next,we obtain the absolute surface shape of the transmissive flat using the liquid transmission method and perform iterative ion-beam polishing of the working surface based on the measured absolute shape to eliminate the effects of gravity and clamping stress.Finally,a stable high-precision transmission surface shape can be achieved. Results and Discussions Finite element models are constructed and numerical simulations are carried out for annular support and three-point support of the mirror frame(Fig.4).Under annular support,the force on the frame is relatively dispersed,so the deformation of both the frame and the flat is smaller,and the deformation of the lower surface of the frame is zero when the mounting datum is an ideal plane.However,when the datum is a rough surface,the deformation of both the frame and the flat changes significantly compared to the ideal case.Under three-point support,the deformation of the frame is larger due to concentrated loading,but changes in the mounting datum have little effect on the deformation of the frame and the flat.Under annular support,the PV value of the residual shape of the working surface,when the transmissive flat is mounted on a standard and rough mounting datum,reaches 8.40 nm,whereas under three-point support,the PV value is only 0.36 nm,which is significantly better.The absolute surface shape of the transmissive flat in a vertical Fizeau interferometer is calibrated using the liquid transmission method to verify the effectiveness of the three-point support structure.The surface shape is calibrated before and after rotation on the same mounting datum surface under both annular and three-point supports(Fig.6).The PV values before and after rotation under annular support are 34.80 and 57.40 nm,respectively,while those under three-point support are 36.64 and 38.59 nm,showing that the working surface is more stable under the three-point support structure.The same transmissive flat is also calibrated on two different interferometers.The calibration results show similar distributions,with PV values around 32 nm and RMS values around 5 nm(Figs.8 and 9),indicating no significant degradation in the surface accuracy. Conclusions The simulation results show that the three-point support structure of the frame is insensitive to variations in the mounting datum surface.Experimental results confirm that the surface precision and profile of the flat remain stable during rotation and cross-system migration under the three-point support structure.Both simulation and experiment indicate that the proposed support structure can effectively suppress the influence of the mounting datum surface and ensure the precision and stability of the transmissive flat in the interferometer.

Objective Ion-beam polishing is a high-precision surface shape-modification technique that requires multiple iterations to ensure surface accuracy. Currently, it is widely used for the surface processing of precision optical flats. However, for transmission flats supported by an adhesive, the temperature generated during polishing may exceed the tolerance temperature of the silicone rubber, which may cause the detachment of the silicone. This changes the uniform support state of the transmission flat, which consequently introduces additional deformation to the surface morphology of the flat. Methods The detachment of adhesive spots occurs primarily in two forms: internal bubbles and flow tendencies. Adhesive spots with bubbles are primarily characterized by a decrease in the contact area with the transmission flat, whereas adhesive spots with flow tendencies primarily exhibit a displacement of the mass center position. We constructed a physical model of a non-uniform support by analyzing the change in force on the transmission flat after debonding. Based on the distance between the adhesive spots and the operating surface of the transmission flat, we classified the adhesive spots into three groups and selected one for analysis. The surface error caused by changes in the adhesive spots was simulated via COMSOL Multiphysics using the finite-element method. The direct result of the finite-element simulation is the global topography of the operating surface of the transmission flat, and the error surface was obtained by subtracting the surface topography under a uniform distribution of the adhesive spots. The results show that the adhesive spots nearest to the operating surface of the flat exert the most significant effect on their surface topography (Figs. 4 and 7). Additionally, we investigated the variation in the low-frequency morphology under different degrees of adhesive-spot detachment. The results indicate that adhesive-spot detachment exerts the most significant effect on astigmatism (Fig. 16). Results and Discussions The liquid-reference method was employed to monitor the iterative ion-beam polishing process of the transmission flat on a Phi 300 mm vertical Fizeau interferometer. The initial peak valley (PV) value of the flat surface before polishing is 122.4 nm. After three rounds of iterative polishing, some adhesive spots around the transmission flat are degummed. An appropriate digital image-processing algorithm was applied to the captured onsite images to calculate the reduction in the area and the displacement of the mass center of the detached adhesive spots. The global surface error was estimated based on the condition of the adhesive spots, and the corrected surface was obtained by subtracting the 300-mm- aperture error surface from the absolute test result via the liquid reference. Further ion-beam polishing was performed based on the corrected surface. Finally, the accuracy of the transmission flat reaches 20.45 nm, which is better than lambda /30, thus validating the effectiveness of the error correction (Fig. 14). The results show that the proposed debonding error-correction method can ensure the convergence of ion-beam polishing (Fig. 15). Conclusions The Phi 300 mm vertical Fizeau interferometer adopted in this study was equipped with a precise temperature control system and an air-floating structure, thus ensuring a constant temperature and vibration isolation in the test environment. Using this device, the repeatability of the liquid surface interference test can reach 0.005 nm (Fig. 17). After assembling the polished transmission flat and considering other error sources, such as detection and light sources, the comprehensive uncertainty of the vertical interferometer is 0.99 nm.

Optical phased array (OPA) is a kind of non-mechanical beam control technology that has extensive application prospects in various fields. However, the initial phase offset of OPA chips will significantly degrade the quality of far-field beams. In this study, a novel hybrid algorithm, to our knowledge, integrating signal reconstruction, rotating-element electric-field vector (REV) techniques, and quantum particle swarm optimization (QPSO) are proposed to address this challenge. Furthermore, our proposed algorithm was employed to calibrate a 512 channel OPA chip. The experimental results demonstrate that, compared with basic particle swarm optimization (BPSO) and deterministic stochastic gradient descent (DSGD), the proposed algorithm effectively escapes local optima and achieves a higher side lobe suppression ratio under comparable iteration counts.

In this study, absolute tests of three flats were performed by using an 800 mm Fizeau interferometer. The flats were installed by employing double-wedge rubber, and the deformation of the flats, primarily involving power and spherical aberrations, was analyzed by conducting finite element analysis. The measurement results revealed an astigmatism component that did not follow the rotation. The astigmatism was separated from the measurement result, and the remaining aberration was consistent with the simulation results, confirming that the astigmatism deformation did not originate from the rubber installation. Subsequently, an absolute test of an 800 mm flat was completed, and the results were compared with those of the traditional three-flat absolute test and Zygo interferometer. The directions of the vertical diameter lines were consistent. The peak-to-valley difference in the full-surface shape was less than 4 nm, and the root mean square was less than 1 nm. The surface errors of flats A and C were consistent with the result of replacing the reference flat, B, with the fourth flat, D, to perform an absolute test. The difference between flats A and C was similar to Zygo's results, thus eliminating the influence of the transmission flat. These results verified the accuracy of the results.

An unstable resonator with seven large aperture ceramic disks and intra-cavity adaptive correction is presented.The composite ceramic disks with absorption rings were adopted to suppress amplified spontaneous emission.An intra-cavity aberration non-conjugate correction based on round-trip wavefront and relaxation iteration was applied in the resonator.After tilt and defocus were corrected in turn,an average output power of 4.5 kW was obtained.The corresponding beam quality factor β was 19.5.After tilt,defocus,and high order aberrations were corrected,the average output power was increased to 5.4 kW,and the beam quality factor β was improved to 6.8.

Liquid mirror can calibrate the interferometer system error by measuring the surface of a steady-state, high-viscosity liquid; however, strict environmental safeguards are typically required. To address this problem, we propose a new liquid mirror method based on liquid micro-amplitude sloshing to calibrate dynamic interferometer system error. According to the multimodal analysis method of fluid finite amplitude sloshing, the liquid surface sloshing surface was modeled and analyzed, a time dimensional mean model was established, and the minimum sampling time was calculated. Finally, the error of the liquid surface sloshing was reduced by time averaging to realize the absolute calibration of the dynamic interferometer's systematic error. According to the proposed minimum sampling time theory, when the mean processing time is greater than the minimum sampling time, the error in the sloshing liquid surface can be controlled to within λ/100. The method’s correctness is proven through an experimental comparison of different calibration methods.

Objective The carrier frequency of the fringes corresponds to is the tilt and constant terms of the phase distribution. Meanwhile, in interferometry, the carrier frequency solution of fringes is significant. It can be adopted for the calibration of phase-shift devices in interferometers and for phase extraction. Second, the carrier frequency parameter is required to correct the retrace errors in interferometers. Additionally, even in the absolute measurement of the two flats, the carrier frequency parameters of the fringes are also employed. Thus, the carrier frequency parameters of the fringes can be utilized in all aspects of interferometry. At present, the carrier frequency parameter solution of fringes can be divided into two categories, with one being the absolute parameter solution method, such as the image processing and Fourier transform methods. This kind of method only employs a single-frame interferogram to compute the absolute value of the carrier frequency, but it has many limitations, including the low computational accuracy of the image processing method and proneness to the singular solution. Meanwhile, the Fourier transform method is only applicable in the case of the high carrier frequency, and cannot be applied to the low spatial frequency interferometric fringes. The Fourier transform method is only applicable to the case of high carrier frequency, and for low spatial frequency interference fringes, its spectrum is coupled with the zero frequency, which is difficult to separate with the large solving error. In response to the limitations of the single-frame method, we carry out the research on the phase-shift method, which is a class of relative parametric solution methods. Its essence is a random tilt phase-shift algorithm, which is mainly adopted for phase solution, but incidentally, the phase-shift between the interferograms or the relative value of the carrier frequency parameter can also be obtained. Methods First, the carrier frequency parameters (f(x), f(y) and f(z)) are estimated. Then, the interference model is approximated by omitting the higher-order terms of the phase. In such conditions, we can construct a linear fit to solve fz. After obtaining fz, we can obtain parameter f(x) by selecting a row of elements and constructing a new fit. Similarly, we can select a column of elements and obtain the parameter f(y). Finally, considering that the estimated carrier frequency parameters have errors, the above process is iterated repeatedly to find the accurate parameter values. Results and Discussions Simulations show that the method is applicable to many cases, and despite the even background distribution of the interferogram, the carrier frequency can be accurately calculated. The maximum error can be better than 0.01 lambda, and the minimum can be up to 0.002 lambda, lambda is 632.8 nm. The obtained carrier frequency is employed in phase-shift interferometry, and the root mean square error of the phase can be up to 0.0002 lambda. In the experiments, the proposed method has a carrier frequency error of less than 0.007 lambda, compared with the phase-shift interferometry. When the calculated carrier frequency is adopted for phase-shift interferometry, there is no significant ripple in the phase. Finally, this method has an error in principle because of the phase omission. In principle, the phase should be satisfied to be much smaller than the carrier frequency. With the ratio of the peak to valley (PV) of the phase to the carrier frequency parameter defined as c, the discussion shows that when c<0. 5, the prerequisite that the phase is "much smaller than" the carrier frequency can be approximately satisfied, and the error of this method is better than 0.025 lambda. When c<0. 25, the accuracy can be further improved to 0.01 lambda. Conclusions A new carrier frequency calculation method of fringes is proposed, and the simulation and experiments show that the method is widely applicable, with the error of the calculated carrier frequency better than 0.01. in almost any case. It is worth noting that in adopting the proposed method, it is necessary to satisfy the prerequisite that the higher-order phases in the interferometric fringes are much smaller than the carrier frequency. Additionally, the discussion shows that for the general case, only 2-3 fringes in the interferogram are needed to realize the accurate carrier frequency solution, and even for some high-precision planar phases, only one fringe is necessary. The proposed method also has a wide range of applications, and the carrier frequency accuracy of the method can fully satisfy the phase solution in the phase-shift interferometry without any obvious ripple in the phase. Compared with the existing methods, our method has the following advantages such as simple iteration, high running efficiency, and applicability to the case of uneven background of the interferogram. Meanwhile, it is a method of calculating the absolute parameters of the carrier frequency of a single-frame interferogram, and the carrier frequency can be accurately calculated for the interferometric fringes of almost any spatial frequency.

Freeform gradient index (F-GRIN) media are increasingly used in optical systems. The existing ray tracing methods for F-GRIN have low calculation efficiency and require manual adjustment of step size, making the design process complex. To address this, we propose an adaptive ray tracing (ART) method for F-GRIN. ART obtains an initial step size based on an index directional derivative and the element's length. During the subsequent transmission process, the step size is adaptively adjusted. The accuracy and speed of ART were verified through multiple comparison tests. ART can reduce manual participation and significantly improve efficiency in the optical design of F-GRIN.

We propose a new surface reconstruction method for the absolute measurement of shift rotation that can obtain a full-aperture surface distribution. Our approach reduces the effect of tilt errors on surface reconstruction by increasing the translation ratio of the test mirror. This method requires the test mirror to be measured once in situ, after translation, and after rotation. Because the known measured wavefront is a nonlinear function of the absolute surface distribution, the function of the surface distribution is linearized by performing a Taylor expansion to achieve fast surface reconstruction. Finally, a linear system of equations is constructed to solve the absolute surface distribution. The coefficient matrix is sparse and can quickly and accurately reconstruct the absolute surface distribution. In the simulation, the root mean square error (RMSE) of the reconstructed surface distribution of the method discussed in this study can reach to 0.012nm. Moreover, our method can significantly reduce the effect of tilt errors compared with typical methods, thereby yielding a new, high-precision solution for absolute measurements in interferometry.

The retrace error is caused by the inconsistency of the optical path that the reference and test beams in the interferometer pass through after reflection through the reference flat, which will affect the accuracy of the measurement results. Dynamic interferometers can measure the phase change of a dynamic test object. When the gradient of the measurement surface changes, the retrace error will change. For this reason, a planar retrace error correction method based on the black-box model is proposed for correcting the retrace error in the dynamic measurement of liquid surface. We establish a linear black-box model of the retrace error in planar measurement, substitute the tilt amount of the measured part into the model, calculate the retrace error and deduct it from the measurement results, and finally improve the accuracy of the measurement results. The retrace error correction experiment of liquid dynamic measurement is carried out in Φ300mm vertical Fizeau interferometer. The theoretical analysis and experimental results show that the method can be quickly and accurately corrected for dynamic interferometry in the retrace error.

Traditionally, the interferometric method has been employed to test the refractive index uniformity of flat optical materials. The method mainly involves two approaches:the overturning method and the transmission method. Both methods require a minimum of three flat samples, including the reference flat, the transmission flat, and the sample flat to be tested. This requirement for multiple flats increases the overall cost of testing. We have proposed a novel method for evaluating the refractive index uniformity of transmission flats, which only requires two flats, offering a significant improvement over existing method. The method is realized by using two surfaces of the transmission flat and one surface of the reference flat, and the sample flat is used as the transmission flat. Based on the even-and-odd functions, we can get the even-odd, odd-even and odd-odd parts of the refractive index uniformity directly. To the even-even part, we resolve it as the even-even-even (eee) and even-even-odd(eeo) part and we can calculate the eeo part by rotate 90 degrees measurement result. Then we resolve the eee part into even-even-even-even(eeee)and even-even-even-odd(eeeo)part and get eeeo part through the rotate 45 degrees measurement result. Finally, by neglecting high-order rotational symmetry terms, we can calculate the refractive index uniformity from the obtained components. This method with its step-by-step decomposition providing an effective and efficient approach to characterize the optical properties of the flat under test. Four random wavefronts are generated as the initial three surface error and the refractive index uniformity errors. The uniformity errors are recovered according to the proposed method. Compared the difference between the recovered results and the initial values, that only 0.6 ppm(10(-6)) in differ. Additionally, the residual error exhibits rotation invariance, which aligns with theoretical expectations. It is important to note that in the low and middle frequency ranges, the residual error is not affected by the surface error of the flat, but solely determined by the refractive index uniformity itself. The theory is verified the experiment on a 100 mm Zygo interferometer and the refractive index uniformity of the same flat is measured by the three flat transmission method. The test flat is made by quartz. The surface error of the two surfaces are better than lambda/10. The flat has a certain wedge angle. The front and the back surface will not generate interference fringes. The reference flat is a microcrystalline flat, better than lambda/10 as well. The result of the two flat method have the same shape with the three flat transmission method. The refractive index uniformity obtained by the two flats methods is only 0.2 ppm different from that obtained by the transmission method, and the peak-valley value is 3 nm different. To ensure accuracy in the evaluation process, error influence is analyzed considering factors such as rotation angle errors and pix offset errors. By evaluating the influence of error, the experimentally verified results illustrate the effectiveness of this method in accurately evaluating the refractive index uniformity of flat optical materials. The proposed method demonstrates promising potential in the evaluation of refractive index uniformity, reducing resource requirements, and improving cost-effectiveness.

A high average power re-frequency operation Fe:ZnSe laser using laser diode side-pumped free-running Er:YAG lasers as activating sources is presented. Two pieces of subsurface layer doped Fe:ZnSe polycrystal are adoptive in a reflective resonator configuration and face-cooled by liquid nitrogen. A maximal Fe:ZnSe laser power of 105 W at a wavelength of 4.1 μm is achieved upon pumping by ten home-made Er:YAG lasers with fiber coupled output working at a frequency of 250 Hz and a pulse duration of ∼420 μs. Corresponding to the maximum Fe:ZnSe laser power, the optical–optical efficiency and slope efficiency with respect to the absorbed pump power are 43% and 44% respectively. The beam quality factor M 2 is measured to be 3.4. To the best of our knowledge, it is the highest output average power of an Fe:ZnSe laser reported.

This study proposed a non-iterative phase tilt interferometry (NIPTI) for the tilt shift calculation and phase extraction, which can solve the random tilt-shift problem caused by external vibration. The method approximates the higher order terms of the phase to make it suitable for linear fitting. Then according to an estimated tilt, the accurate tilt shift can be obtained by least squares method without iteration, and the phase distribution can be calculated by it. The simulation results indicated that the root mean square error of the phase calculated by the NIPTI can reach up to 0.0002λ. The experimental results showed that the calculated phase exhibited no significant ripple when using the NIPTI for cavity measurements in a time-domain phase shift Fizeau interferometer. Moreover, the root mean square repeatability of the calculated phase reached up to 0.0006λ. It indicates that the NIPTI provides an efficient and high-precision solution for random tilt-shift interferometry under vibration.