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Abstract
Ultra-precise reflectors in the advanced light source facilities urgently call for local slope error measurements with nano-radian precision. However, the existing methods currently utilized in the long trace profiler systems struggle to meet the requirements. In this paper, we present a weak-value amplification enhanced absolute local slope measurement scheme, in which the surface height difference between two adjacent points can be measured directly with precision on the pico-meter level. As a result, the absolute local slope measurement reaches a record precision level of 9.7 nrad (RMS) with a small lateral separation of 0.5 mm. Comparing to the existing methods, our scheme is more disturbance-resistant, more compact and cost-effective. The local curvature measuring capability is also validated with two synchronously parallel local slope measurement paths, between which the separation is set as 2mm. A local curvature measurement is obtained with precision of 3.4 × 10−6m−1 (RMS) and its corresponding slope variation is 6.8 nrad. Our method exhibits important application prospects in the field of ultra-precise surface fabrication inspection.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Advanced light source facilities such as synchrotron radiation and X-ray free electron lasers are playing imperative roles in the development of modern science. X-ray reflectors, such as Kirkpatric-Baez optics (KB optics), are widely used for focusing or collimating the beams of those light sources [1–3]. As the working wavelength range of these light sources under construction trying to reach sub-nanometer scale, it is necessary to measure the local slope error of reflector surfaces with a precision on the nano-radian level, which becomes a huge challenge to all the prior arts.
At present, the local slope measurement (LSM) methods can be generally categorized into three mechanisms by geometrical features.
The autocollimator with a Fourier Transform (FT) lens of meter-level focal length is a typical example of the first type (Fig. 1(a)) and prevalently utilized in the existing long trace profiler (LTP) systems. Though the LSM precision can be improved by increasing the focal length of FT lens in theory, the measurement uncertainty brought by the atmospheric turbulence and the temperature as the light path extends and the deterioration of the repeatability of spot displacement due to the larger spot size on the detector hinders the improvement in practice. Many research groups thus seek to improve the LSM precision by developing algorithms to estimate the spot location more accurately. To date, the highest LSM precision reported is 20 nrad [7–9]. Another commonly used mechanism is interferometry. There are two main schemes for such systems, as shown in Fig. 1(b) and (c). One famous example of the mechanism in Fig. 1(b) is the Fizeau interferometer, which requires a reference surface. The precision of the measurement is influenced by the manufacture precision of the reference. Moreover, for each device under test, two measurements are needed, leading to a higher systemic error. The lateral shearing interferometer is the typical application of the third mechanism, which is an absolute measurement method and thus reduces the effect brought by the noise induced by the photoelectric detection and the environment disturbance greatly. However, to achieve an LSM precision on the nano-radian level, the required measurement repeatability of the height difference, namely the optical path difference, is on the pico-meter level or less, which brings a huge challenge to the phase measurement.
Fig. 1. Schematics of three types of LSM methods. (a) By measuring the lateral displacement of the light spot location $\Delta x$ that the reflected pencil beam striking on a certain plane [4], we can calculate the local slope through $\kappa = \Delta x/2f$, in which f is the focal length of the FT lens. (b) By measuring the heights of two adjacent points of the surface [5], we can calculate the local slope from the difference of two heights, which can be expressed as $\kappa = ({{h_2} - {h_1}} )/\mathrm{\Delta }r$. (c) By measuring the height difference directly between two adjacent points [6], we can calculate the local slope via $\kappa = \mathrm{\Delta }h/\mathrm{\Delta }r$.
Weak value amplification (WVA) is inspired by quantum weak measurement to measure a tiny physical quantity change [10–21]. WVA exhibits precisions that exceed traditional methods to a great extent in many applications, such as optical phase estimation [22–26], and beam deflection measurement [27,28]. Dixon et al. demonstrates the beam deflection measurement with angle resolution of 400 frad by incorporating the WVA effect in a rectangularly arranged Sagnac interferometer [27]. However, the deflection results of their scheme would be sensitive to the vibrations of the device under test in the vertical direction, therefore impairing the repeatability and the precision of the measurement.
In this work, we present a WVA-enhanced absolute local slope measurement (WALSM) method. We incorporate WVA into the shear differential local slope measurement and acquire an ultrahigh precision on the nanoradian level over a sub-millimeter-scale local region with a compact system setup. By scanning over a surface area, we can reconstruct the local slope information of the X-ray reflector surfaces and monitor the manufacturing precision. Moreover, due to the common-light-path feature of the measurement, the phase difference is only determined by the slope. As a result, it is resistant to the environment disturbance and insensitive to the piston-type vibration. Furthermore, by measuring the local slopes of two adjacent position synchronously, we can calculate the curvature of the surface, possessing the potential to reconstruct the surface morphology more accurately.
2. Theoretical analysis
As shown in Fig. 2, the WALSM system includes three sections: pre-selection, weak coupling and post-selection. Two orthogonal polarizers are used as the pre and post selection respectively to obtain the suitable polarization states. The meter denotes the transverse position distribution of the beam and the pointer denotes the expectation of the position. Inside the red dashed box is a lateral shear differential optical path, where the weak coupling occurs. A phase difference $\phi $ depending on the slope $\kappa $ is thus acquired.
When an incident light beam of arbitrary polarization state enters the system, the pre-selection changes it into the initial state $|{{\psi_i}}\rangle $, and the joint state of the system and the meter is $|\mathrm{\Phi }\rangle = \int dq\varphi (q )|q \rangle|{\psi _i}\rangle $, where q is the transverse position of the meter, and $\varphi (q )$ is the wave function of the meter in the coordinate representation. The designed quantum system ($\hat{A}$) is a two-level system and the left and right beam paths are denoted with $\{{|{0\rangle ,{\; }} |1\rangle } \}$, respectively. Thus, the corresponding initial state is $|{{\psi_i}}\rangle = ({|0\rangle + |1\rangle } )/\sqrt 2 $.
In the WALSM, the incident beam is of Gaussian distribution. At the distance z corresponding to the waist, the meter is written as
Fig. 3. (a) Front view of the lateral shear differential part, the displacement between the beams is d$= 0.5\textrm{mm}$. Since the slope of the reflector is slight, there is only phase difference between the beams. In yaw direction, there is almost no rotation. (b) Left view of the lateral shear differential part. The reflector is set in x-y plane, the angle between its normal direction and the x axis is $\theta = 8.7 \times {10^{ - 3}}\; \textrm{rad}$.
Therefore, the paths in the BD of the reflected beams slightly deviate from the incident beams, leading to a separation $\Delta q$ between the $|0\rangle $ and $|1\rangle $ beams in the z direction.
where $\mathrm{\alpha }$ is the refractive index modified coefficient of the BD. It can be estimated as $2.31\; \textrm{ra}{\textrm{d}^{ - 1}}$ when $\theta $ is small. Therefore, the coupling between the which-path observables and the transverse positions can be achieved. With this effect, the system and meter are entangled, which leads to an evolution operator in the form of $\textrm{exp}({ - ig\hat{A}\hat{p}} )$, where $g = \mathrm{\Delta }q/2$ is the relative offset of the meter in the coordinate representation, p is the transverse momentum of the beam. The system operator $\hat{A}$ is $|0 \rangle \langle 0|{ - |1 \rangle \langle 1} |$, whose eigen value is ${a_i} ={\pm} 1$. After the system and meter are coupled by BD and the reflector, the joint state evolves intoIn the limited area to be measured, we can take the reflector as a plane of a slope $\kappa $ corresponding to z axis, and the phase difference of the normally incident $|0\rangle $ and $|1\rangle $ is $\phi = 4\pi d\kappa /\lambda $. After passing through the post-selection, the system collapses to the final state. The state of the post selection is $|{{\psi_f}}\rangle = ({{e^{ - i({\phi /2} )}}|0\rangle - {e^{i({\phi /2} )}}|1\rangle } )/\sqrt 2 $. As a result, the weak value ${A_w}$ can be written as
Since $\theta $ is small, the pointer shift is greatly amplified by the post-selection. We can calculate the local slope of the reflector by monitoring the amplified pointer shift. By simply displacing the incident beam into two synchronous ones, we can measure the local slope of two adjacent position synchronously and thus calculate the curvature.
3. Experimental setup
The slope/curvature measurement system setup is shown in Fig. 4. The randomly polarized light comes out of the He-Ne laser (JDSU, 1145, 22.5 mW), then is coupled into the WALSM via a single-mode fiber collimator (FC). After passing through the first beam displacer (BD1), the light beam is divided into two parallel beams with a separation of 2 mm in the x-z plane. Each beam leads a slope measurement system, which synchronizes with the other one. Polarizer (P1) acts as the pre-selection, fabricating the two beams into pre-selected states $|{{\psi_i}}\rangle $ simultaneously. The 50/50 beam splitter (BS) placed with a 45$^\circ $ angle between the x-z plane is influence-free to the meter and system. The second beam displacer (BD) divides each beam into parallel beams with a separation of 0.5 mm in the x-z plane.
Fig. 4. Schematic of the slope/curvature measurement system. The curvature measurement is realized by the synergy of a two-path WALSM system. The red solid line denotes the light path of the first WALSM system, and the red dotted line denotes the light path of the second WALSM system. The two paths are laterally displaced by a beam displacer. Finally, the images are captured simultaneously by a CCD camera.
We choose a single crystalline silicon chip with the dimension of 77${\times} $11.8${\times} $0.5 mm as the target deformable reflector (SR). One edge of the SR is fixed and we use a piezo-motion driver (close-loop control with 10 nm accuracy) to push the opposite edge pivoting the z-axis to create a deformation. The reflected beams and the incident beams are both in the x-y plane, with an included angle of 2$\theta $. Under the tiny-deflection approximation, the deflection of the reflector is
where $g(r )= {r^2}({L - r} )/2{L^3}$, h is the displacement caused by the driver, $L = 75\; \textrm{mm}$ is the distance between the fixed point and the driving point, and r is the distance between the measuring point and the fixed point. From Eq. (6), we can see that the slope of the reflector is $\kappa (r )= g^{\prime}(r )h$, which is proportional to the displacement.Then the two beams merge together at BD. After being reflected by BS, considering the indistinction error of the phase difference induced by the imperfection of the optical components, the beams first enter a phase compensator consisted of a quarter wave plate (QWP) and a half wave plate (HWP), and thus enable the WVA system to work within the ideal range. Then it enters a -45$^\circ $ placed polarizer (P2) for post-selection. Eventually, the transverse distribution of the beams is monitored by a CCD camera (FILR, GS3-U3-60QS6M-C).
4. Local slope measurement
The precision of the WALSM system is validated by measuring the local slope change of a deformed silicon reflector. In experiment, we choose two locations ($r = 3.85\; \textrm{mm},5.85\; \textrm{mm}$) to monitor the slope changes. The fixed tilting angle of the reflector in the x-y plane $\theta $ is ${0.5^ \circ }$. The region of interest (ROI) of the CCD is set to be 800${\times} $900 pixels, and the position z is $0.44\; \textrm{m}$. Every five measurements are averaged as the measurement result, which does not significantly reduce the measurement efficiency. Considering the nonlinear effect of WVA, the sensitivity will lower the dynamic range [31]. Certain compromises are inevitable according to actual measurement requirements. We set the piezo driver to be 9 steps, and the displacement of each step is 100 nm. Within each step, the uncertainty is estimated by 25 trial results. The pointer shift corresponding to the movement of the piezo driver is shown in Fig. 5(a).
Fig. 5. (a) The pointer shift corresponding to the piezo driver shift. The curves with error bars are the measured pointer shift. The solid lines are linear fits of the measured data. The insets above the red line are the images of the beams collected by the CCD in steps 2, 5, and 8 respectively. Inset at the lower right corner: the total shift data collected in the first step. (b) The experimental and theoretical relationships between the pointer shift and the piezo driver shift at $r = 3.85\; \textrm{mm}$. Inset: the amplified pointer shifts in step one and two. As it is shown, the shift between steps can be easily distinguished.
It can be seen that the pointer shift is linearly proportional to the movement of the piezo driver, i.e., linearly proportional to the slope in the range of $0\sim 1.5\; \mathrm{\mu} \textrm{rad}$. The deviation between the experimental data and the theoretical curve is mainly induced by the piezo-motion driver. The inset in the lower right corner shows all the centroid data of the first step, from which we can see that the disturbance of the two WALSM systems are almost synchronized. In Fig. 5(b), we take the first measured data (red) as the calibration reference then plot the pointer shift and the theoretical slope curve, which are consistent with each other within acceptable error range.
We define the pointer shift per unit slope as the sensitivity
In our WALSM system, the sensitivity ${s_w}$ is $4z/\alpha \theta $, which is linearly proportional to the distance between the beam waist and the detector and inversely proportional to the tilting angle. Therefore, the sensitivity can be increased by both lengthening z and decreasing $\theta $. In experiment, our system exhibits a sensitivity of 87.8 nm/nrad, with a measurement uncertainty of 9.7 nrad (RMS) theoretical dynamic range of 4.5$\mu \textrm{rad}$. In specific application scenarios, the dynamic range can be further enlarged by incorporating techniques such as nonlinear correspondence rules and phase-compensator modulation. Compared to commonly used autocollimator with the FT lens focus length of 1 meter, whose sensitivity is $2\; \textrm{nm}/\textrm{nrad}$, our sensitivity shows a 43.9-fold increasement. Moreover, since the dimensions of the two systems are mainly determined by z and f, correspondingly, our system is more compact.
In general, during the slope measurement, the main sources of the uncertainty are vibration, airflow, stray light and the noise of the detector. Compared to traditional surface profiling methods using the interferometers, the WALSM system is almost immune from the vibration along the x axis, as well as the vibrations in roll and pitch directions. The error of stray light induced by surface defects and multiple reflection of the optical components are stable since the configuration of the system is remained unchanged during measurements. The major influence on the slope measurement error is the biased estimation of the pointer shift, which is a systematic error and the repeatability is kept unaffected. The noise brought by the CCD detector is inevitable. WVA weakens the detected light intensity and thus brings up the noise, which is one of the major sources of the uncertainty. At the same time, the effect of the airflow can only be lowered but not eliminated. Therefore, there is a phase difference between $|0 \rangle , |1\rangle $. Since the airflow disturbance is time variant, the corresponding error is random. Besides, the measurement uncertainty of the beam’s centroid can go up to $0.86\; \mathrm{\mu} \textrm{m}$ (0.19 pixel), which is not ideal. It can be furthered suppressed through existing technology such as cooling method, and reach 0.01 pixel. Which is to say, the precision of the slope measurement can be further improved.
5. Local curvature measurement
Curvature can give us surface information on a higher order, enabling a more accurate surface reconstruction. Moreover, since the curvature is the inherent property of the reflector, the influence of the positioning posture of the reflector is much smaller than that on the slope. Also, the curvature measurement suffers less from the environment vibration to a large extend. For cylindrical surface with one-dimensional deformation, the curvature is
For our reflector, $f(r )$ can be simplified as a one-dimensional function in z direction. From Eq. (6), we can get that $K(r )= |{g^{\prime\prime}(r )h} |{({1 + {\kappa^2}} )^{ - 3/2}}$. In experiments, $\kappa (r )$ can be taken as an infinitesimal and thus the curvature of the reflector can be written as
By constructing two synchronous WALSM systems, we can measure the one-dimensional curvature. The curvature can be expressed as
where ${q_1}$ and ${q_2}$ denotes the pointer shifts of the two WALSM systems, correspondingly. The schematic of the curvature measurement system (CMS) is shown in Fig. 4. In our system, we place a beam displacer in front of the pre-selection and divide the beam into two parallel beams with a separation of ${d_0}$=2 mm between each other. With two synchronous WALSM systems, we can acquire the curvature of the reflector. The experimental curvature measurement result of the reflector is shown in Fig. 6.Fig. 6. The experimental curvature measurement result of the reflector. The blue dots with error bars are the measured data. The red curve is the linearly fitted result. Inset: the pointer shift collected in the first step.
It can be seen that the pointer shift sensitivity of CMS can reach $43.1\; \textrm{m}/{\textrm{m}^{ - 1}}$, with a theoretical dynamic range of $2.25 \times {10^{ - 3}}{m^{ - 1}}$. The uncertainty is $3.4 \times {10^{ - 6}}{\; }{\textrm{m}^{ - 1}}$, equivalent to the curvature radius of $2.96 \times {10^5}\textrm{m}$ (RMS), corresponding slope variation is 6.8 nrad,. The realization of the curvature measurement is via two synchronous WALSM systems. The vibration in the yaw direction is also synchronous and the influence can be greatly alleviated. The large-scale slow-variant environment influences, such as stable airflow disturbance and environment temperature, can be viewed as constant in the scale considered, and the influences on the two WALSM systems are almost identical. Consequently, the CMS can reduce the low-frequency environment disturbance. The main source of uncertainty is the noise of the CCD. The uncertainty of the pointer shifts of the WALSM systems is negatively correlated to the number of photons, and positively correlated to the electronic noise of the detector. For our CMS with dual WALSM systems, the pointer shift variance is $VAR({\langle {q_2}\rangle - \langle {q_1}\rangle } )= 2VAR({\langle q\rangle } )$, indicating that the variance induced by CCD noise of the CMS doubles that of one single WALSM system. And uncertainty is the root of variance. However, the measured uncertainty of the pointer shift of CMS is $0.58\; \mathrm{\mu} \textrm{m}$ (0.13 pixel) which is even smaller than a single WALSM system. That means our system can suppress the systematic disturbance and therefore increase the precision. Through the experiment, the environment perturbation is not strictly controlled and the signal-to-noise ratio (SNR) of the CCD used is mediocre. If we can further improve the environment through air tightness and strict temperature control, as well as use detectors with higher SNR, we will be able to acquire higher measurement precision.
6. Conclusion
In this work, an absolute local slope measurement scheme-WALSM- is presented. Compared to commonly-used autocollimators, the WALSM is able to increase the sensitivity by 43.9 folds with a more compact setup. Considering practical application, with a careful tradeoff in the system design, we acquired a sensitivity of 87.8 nm/nrad. The precision of local slope measurement can reach 9.7 nrad RMS over a region of 0.5 mm. By simply inserting a beam displacer into the light path, we can construct two synchronous WALSM systems and measure the curvature information. The precision of local curvature measurement is $3.4 \times {10^{ - 6}}\; {\textrm{m}^{ - 1}}$ RMS. Our WALSM system is insensitive to piston-type vibration when measuring local slope and shows a good common mode suppression effect on low-frequency environmental disturbances. It shows the superiority of WALSM scheme in ultra-precision surface profiling measurement, and the precision can be further improved by controlling the environment and increasing the SNR of the CCD used.
Acknowledgement
We appreciate the helpful discussions and advices from Professor Han Sen (University of Shanghai for Science and Technology, USST) during our paper preparation.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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