Ultra-sensitive angle sensor based on laser autocollimation for measurement of stage tilt motions

Yuki Shimizu; Siew Leng Tan; Dai Murata; Taiji Maruyama; So Ito; Yuan-Liu Chen; Wei Gao

doi:10.1364/OE.24.002788

1. Introduction

Precision positioning technology is essential in various industrial fields such as semiconductor, precision machining or precision measurement. As fundamental mechanical components for precision positioning, linear stages [1] or planar motion stages [2, 3] are often employed. Closed-loop feedback control of the stage systems with high precision displacement sensors such as laser interferometers [4] or linear encoders [5] having sub-nanometric measurement resolution has enabled to achieve nanometric positioning accuracy.

In general, precision positioning systems have six-degree-of-freedom motion error components being categorized as in-plane motion error and out-of-plane motion error. For example, in the case of a linear stage having the X-axis as a primary axis as shown in Fig. 1, the in-plane motion error of a moving table consists of two-directional translational motion error components (ΔX and ΔY) and a tilt motion error component (Δθ_Z), while its out-of-plane motion error consists of one-directional translational motion error component (ΔZ) and two-directional tilt motion error components (Δθ_X and Δθ_Y). Among these error components, ΔX, which is the difference between the command position and the actual position of the moving table along the primary axis, is controlled by a closed-loop control with a position sensor such as a laser interferometer or a linear encoder. A measurement axis of the position sensor is required to be aligned coaxially with the primary axis of the moving table so that the Abbe error [6] caused by the tilt motion error components of the moving table can be avoided [7]. Some of the state-of-the art positioning systems, such as a Multi-scale Alignment and Positioning System (MAPS) [8] or Nanopositioning and Nanomeasuring Machine (NMM) [9], have therefore been designed in complying with the Abbe principle, and have achieved sub-nanometric positioning accuracy.

Fig. 1 A schematic of motion error components of a precision linear stage.

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However, on the other hand, many other positioning systems such as those for machine tools and coordinate measuring machines (CMMs) do not satisfy the Abbe principle due to the restrictions on their geometric designs [10]. Even a small tilt motion error component of a moving table causes a huge Abbe error; for example, when the distance between the measurement axis of a positioning sensor and the primary axis of a moving table is 100 mm, an angular motion error component Δθ_Z of 1 arc-second results in the translational motion error component along the primary axis ΔX of 0.48 μm [11]. The tilt motion error components are therefore required to be reduced as much as possible by improving the machining accuracy of mechanical components in a positioning system. Especially, surface form accuracy of guideways in the positioning system is of great importance since it strongly affects the tilt motion error of the moving table. Even the guideways can be machined to a certain level of accuracy on the order of 5 to 10 μm [12], their surface form accuracy can easily be affected during the assembly process of the stage system. Both error verification and error correction processes are therefore required to be implemented to the fabrication process of ultra-precision positioning systems. To carry out the correction process, tilt motion error components are required to be measured many times by using measurement instruments. Especially, for the achievement of nanometric or sub-nanometric positioning accuracy, tilt motion error components of a stage system are required to be measured with a resolution of better than 0.001 arc-second.

Laser interferometers with multiple measurement laser beams or optical angle sensors based on laser autocollimation method [13], which are referred to as autocollimators, are commonly used to measure tilt error motion components of a linear stage [4, 14]. However, it is difficult to apply the laser interferometers for measurement during the error correction process of the guideways since the path of the laser beam will often be blocked by operators for the error correction. In addition, the measurement resolution of the laser interferometers is limited to be up to 0.04 arc-second [4]. Therefore, a commercially available autocollimator, which can continue its measurement even if its optical path is interrupted by the operators, is often employed in the error correction process of the guideways.

An autocollimator employing a collimator objective with a long focal length and a charge coupled device (CCD) as a position-sensing detector can achieve the measurement resolution of up to 0.005 arc-second [14]. Furthermore, there also exists a commercial product achieving high measurement resolution of up to 0.002 arc-second by employing a dual axis analog silicon-based sensor [15]. However, on the other hand, such autocollimators tend to be bulky; for example, the total length of the optical sensor head for the above mentioned autocollimator achieving the resolution of 0.002 arc-second is more than 270 mm. It is sometimes difficult to employ such large autocollimators in a factory line of ultra-precision linear stages, in which a compactness of the measurement instrument is of great importance since it should be placed in a limited space of an assembly line. In addition, it is still difficult for such autocollimators to achieve measurement resolution better than 0.001 arc-second.

A highly-sensitive angle sensor with a high response speed has therefore been developed based on the laser autocollimation method [16]. A quadrant photodiode (QPD) has been employed as a light position detector to achieve higher response speed. Throughout some experiments, it has been revealed that measurement sensitivity of the developed angle sensor does not depend on a focal length of a collimator objective, and the developed two-dimensional angle sensor has been verified to have the measurement resolution of 0.01 arc-second, while achieving the response speed of better than 1 kHz. In addition, it has also been revealed that a smaller diameter of the focused light spot on the photosensitive surface of the QPD (QPD plane) is effective in improving the measurement resolution of the angle sensor [17]. Meanwhile, it is still difficult to achieve further higher measurement resolution since most part of the energy in the focused light spot will be lost at insensitive areas (gaps) in-between active areas on the QPD plane, resulting in a degradation of a signal to noise ratio (S/N) of the sensor output [18]. This problem can be solved by a new optical configuration having single-cell photodiodes (SPDs), which can accept the focused light spot with a further smaller diameter. In this optical design to be addressed in the next section of this paper, an increase of the diameter of the measurement laser beam, which is made incident to the collimator objective, is effective in reducing the focused light spot diameter on the photosensitive surface of the SPD (SPD plane). Meanwhile, the measurement laser beam with a larger diameter is at a risk of being affected by aberrations of a collimator objective, which influences the light intensity distribution of the focused light spot and degrades the angle sensor sensitivity. However, in the design concept of the angle sensor with the SPDs, attentions have never been paid for the light intensity distribution of the focused light spot, whose diameter is ranging from several μm to several-ten μm. For the achievement of the measurement resolution better than 0.001 arc-second, systematic approaches from both the computer simulation and the experiments are therefore required to be carried out.

In this paper, computer simulation based on wave optics is firstly carried out. An optical model, in which the influence of the spherical aberration of the collimator objective in the angle sensor is implemented, has been established to estimate the sensor sensitivity. To verify the feasibility of the computer simulation, experiments have been carried out by using an optical setup built on an optical bench. Furthermore, a prototype optical angle sensor based on the proposed concept has also been developed, and its measurement resolution has been verified in experiments.

2. Principle of the ultra-sensitive optical angle sensor

The ultra-sensitive optical angle sensor developed in this paper is based on laser autocollimation [13], in which tilt motions of a reflector mounted on a measurement target such as a linear stage will be detected as displacements of a focused light spot on a photosensitive plane of a light position detector. A schematic of the optical configuration for the two-dimensional ultra-sensitive optical angle sensor employing SPDs is shown in Fig. 2. The optical angle sensor consists of the reflector and an optical sensor head, in which a laser diode (LD) and single-cell photodiodes (SPD1, SPD2 and SPD3) are employed as a light source and light position detectors, respectively. A diverging laser beam from the LD is collimated by using a collimating lens (CL) so that the collimated laser beam can be employed as the measurement laser beam. In the optical sensor head, the measurement laser beam reflected from the reflector surface is divided into two laser beams by using a beam splitter (BS1). One of the laser beams is made incident to a single-cell photodiodes (SPD3) so that intensity fluctuation of the light source can be monitored. Meanwhile, the other laser beam is divided into two beams again by using another beam splitter (BS2), and the generated laser beams are made to focus on the SPD1 and SPD2 planes through the collimator objectives CO1 and CO2, respectively. The SPD1(2) in combination with CO1(2) is used as an autocollimation unit (θ_Y₍_Z₎-unit).

Fig. 2 A schematic of the optical configuration for the ultra-sensitive optical angle sensor employing single-cell photodiodes (SPDs) (A laser beam with a high truncation ratio is used as the measurement laser beam).

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In the θ_Y₍_Z₎-unit, the focused light spot is positioned to be on the edge of the active area on the SPD1(2) plane to detect the light spot displacement Δu(v) due to the tilt motion Δθ_Y₍_Z₎. According to the geometric optics, a relationship between Δθ_Y₍_Z₎ and Δu(v) can be described by the following equation:

\begin{array}{l} Δ u (v) = f \cdot \tan 2 θ_{Y (Z)} \\ \approx 2 f Δ θ_{Y (Z)} \end{array}

where f is the focal length of the CO1(2). Now we assume that the center of the focused light spot is positioned on the edge of the active area on the SPD1(2) plane when Δθ_Y₍_Z₎ = 0. On the assumption that the edges of the active areas on SPDs are perfect, denoting the intensity distributions of the focused light spots on the SPD1 and SPD2 planes by I₁(u,v) and I₂(u,v), respectively, outputs of the angle sensor u_{SPD_out} [%] and v_{SPD_out} [%] can be calculated by the following equations:

u_{SPD_out} = \frac{\iint_{S 1} I_{1} (u, v) d u d v}{\iint_{S 1_total} I_{1} (u, v) d u d v} \times 100

v_{SPD_out} = \frac{\iint_{S 2} I_{2} (u, v) d u d v}{\iint_{S 2_total} I_{2} (u, v) d u d v} \times 100

where S1 and S2 mean the regions of the active areas on the SPD1 and SPD2 planes illuminated by each focused light spot, respectively, while S_{1_total} and S_{2_total} mean the whole areas on the SPD1 and SPD2 planes illuminated by each focused laser beam, respectively. The numerators in Eqs. (2) and (3) can be acquired as photocurrent outputs from the SPD1 and SPD2, respectively. Meanwhile, the denominators in Eqs. (2) and (3) are equal to a half of the laser beam power accumulated by the SPD3. By monitoring the variation of the photocurrents from the SPD1, SPD2 and SPD3, two-degree-of-freedom tilt motions of the reflector can therefore be measured simultaneously. In addition, by employing a scale grating as the reflector and capturing 1st-order diffracted beam by an additional autocollimation unit, simultaneous three-degree-of-freedom tilt motion measurement can also be carried out [11].

On the assumption that the light intensities I₁(u,v) and I₂(u,v) are uniform, Eqs. (2) and (3) can be simplified as follows:

u_{SPD_out} = \frac{\frac{1}{2} π {(d_{1} / 2)}^{2} + d_{1} \cdot Δ u}{π {(d_{1} / 2)}^{2}} \times 100 = 50 + \frac{8 f Δ θ_{Y}}{π d_{1}} \times 100

v_{SPD_out} = \frac{\frac{1}{2} π {(d_{2} / 2)}^{2} + d_{2} \cdot Δ v}{π {(d_{2} / 2)}^{2}} \times 100 = 50 + \frac{8 f Δ θ_{Z}}{π d_{2}} \times 100

where d₁ and d₂ are the diameters of the focused light spot on the SPD1 and SPD2 planes, respectively. Differing from the optical angle sensors employing QPDs, which have insensitive gaps in-between four active cells on their photosensitive surfaces, there is no restriction on the acceptable focused light spot diameters d₁ and d₂ for the optical angle sensor with SPDs. The sensor sensitivity can therefore be maximized by minimizing the focused light spot diameters d₁ and d₂ to the light diffraction limit. As a result, the optical configuration is expected to achieve higher measurement resolution that cannot be achieved by the conventional angle sensor with the QPDs. It should be noted that, in practical case, the focused light spots have non-uniform intensity distribution on the SPD planes.

Now we focus on the focused light spot diameters on the SPD planes. When the collimator objective is free from aberrations, the focused light spot diameter d_diff on the focal plane of the collimator objective will be determined by the diffraction limit as shown in Fig. 3(a), and can be expressed by the following equation [19]:

d_{diff} = \frac{2.44 f λ}{D}

where D and λ are a diameter and a wavelength of the measurement laser beam, respectively. The focal plane of an aberration-free system is often referred to as a paraxial focal plane [19]. Meanwhile, in practice, the collimator objective is not free from the aberrations. As a result, the focused light spot diameter at the paraxial focal plane will be affected by the aberrations; especially, in the case of the optical angle sensor, the spherical aberration is one of the most dominant factors. As can be seen in Fig. 3(b), when D becomes large, a marginal ray strikes the position with the displacement of d_TA from the optical axis on the paraxial focal plane due to the influence of the spherical aberration. The displacement d_TA will be determined by the Gaussian optics, and can be described by the following equation [20]:

d_{T A} = \frac{D^{3}}{32 f^{2} {(n - 1)}^{2}} [n^{2} - (2 n + 1) \frac{R_{2}}{R_{2} - R_{1}} + \frac{n + 2}{n} {(\frac{R_{2}}{R_{2} - R_{1}})}^{2}]

where n, R₁ and R₂ are a refractive index and surface radii of the collimator objective, respectively. As can be seen in Eq. (7), the influence of the spherical aberration will be increased with the increase of D. It should be noted that the physical meaning of d_diff (Airy disk diameter) in Eq. (6) is different from that of d_TA. According to Eqs. (6) and (7), the increase of D is effective in reducing d_diff for the achievement of a highly-sensitive angle sensor. Meanwhile, the influence of the spherical aberration, on which attentions have never been paid so far in terms of the optical angle sensor, needs to be taken into consideration in the case of the measurement laser beam having a larger D. In the following section, computer simulation based on wave optics is therefore carried out to investigate the influence of the spherical aberration on the sensitivity of the optical angle sensor.

Fig. 3 An influence of the spherical aberration of the collimator objective on the focused light spot. (a) without the influence of the spherical aberration; (b) with the influence of the spherical aberration.

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3. Simulation on the angle sensor sensitivity

As described in the previous section, the sensitivity of the optical angle sensor can be increased by decreasing the focused light spot diameter on the SPD plane. For the decrease of the focused light spot diameter, the increase of the measurement beam diameter D is effective, while the influence of the spherical aberration of the collimator objective needs to be taken into consideration. The spherical aberration not only increases the focused light spot diameter, but also causes a disturbance in its intensity distribution, which greatly influences the sensitivity of the optical angle sensor. Therefore, in this section, computer simulation is carried out based on wave optics to investigate the influences of the spherical aberration on the sensor sensitivity.

Figure 4 shows a schematic of an optical model simulating an autocollimation unit in the optical angle sensor. In the optical model, a plano-convex lens is employed as the collimator objective in the autocollimation unit, while a pupil plane and the SPD plane are placed at its front focal position and back focal position, respectively. The pupil plane having an aperture diameter of D is included in this optical model so that the corresponding measurement beam diameter D can be adjusted. The intensity distribution of the focused light spot I₁(u,v) on the SPD plane can be described by the following equation [21]:

I_{1} (u, v) = \frac{1}{λ^{2} f^{2}} {| \iint U_{l} (x, y) P (x + u, y + v) \exp [- j \frac{2 π}{λ f} (x u + y v)] d x d y |}^{2}

where P(x,y) and U_l(x,y) are a pupil function and a complex field across the pupil plane. The pupil function P(x,y) is defined as follows:

P (x, y) = {\begin{matrix} 1 (w h e n ρ (x, y) < 1) \\ 0 (w h e n 1 < ρ (x, y)) \end{matrix}

where ρ(x,y) is the normalized radial coordinate, and can be described as follows:

ρ (x, y) = \frac{\sqrt{x^{2} + y^{2}}}{D / 2}

Once the intensity distribution I(u,v) is calculated from Eq. (8), the optical angle sensor outputs can be calculated from Eqs. (4) and (5). In the optical model, the pupil plane is assumed to be illuminated by a plane wave with amplitude of A uniform across the surface. The influence of the spherical aberration of the collimator objective can be expressed as wavefront error, and can implemented in the calculation by using the complex field U_l(x,y). In the following calculations, U_l(x,y) represented by the following equations is applied to Eq. (8) [20, 21].

U_{l} (x, y) = {\begin{matrix} A (without aberration) \\ A \cdot \exp [j \frac{2 π}{λ} S ρ^{4} (x, y)] (with aberration, at paraxial focus) \\ A \cdot \exp [j \frac{2 π}{λ} S (ρ^{4} (x, y) - ρ^{2} (x, y))] (with aberration, at best focus) \end{matrix}

where the parameter S is referred to as the peak aberration coefficient. In the case of a plano-convex lens, S can be described by the following equation [20, 22]:

S = \frac{- [n^{3} + (n + 2) + (3 n + 2) {(n - 1)}^{2} - 4 (n^{2} - 1)] \cdot D^{4}}{32 n {(n - 1)}^{2} f^{3}}

Parameters employed in the following calculations are summarized in Table 1. These parameters are defined in consideration of the experiments, the details of which are described in the next section. Figure 5 shows examples of the U_l(x,y) calculated by using the parameters in Table 1. The wavefront errors are represented in a wave number. As can be seen in Fig. 5(b), the influence of the spherical aberration is expected to become significant near the edge of the projected aperture diameter.

Fig. 4 An optical configuration for computer simulation based on wave optics.

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Table 1. Parameters used in the computer simulation.

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Fig. 5 Wavefront errors due to the spherical aberration of the collimator objective applied to the simulation model (results in the case of D = 14 mm). (a) without the spherical aberration; (b) with the spherical aberration, at paraxial focal plane; (c) with the spherical aberration, at best focal plane.

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As the first step of the computer simulation, intensity distribution I₁(u,v) of the focused light spot at each measurement laser beam diameter D is calculated. Figures 6(a) and 6(b) show the calculated intensity distributions with D of 2 mm and 14 mm, respectively, and Fig. 6(c) shows their cross sections. In the figures, intensity distributions normalized by the maximum intensity calculated with the aberration-free system are plotted. In the case of small D ( = 2 mm), the influence of the spherical aberration is negligibly small. However, on the other hand, the influence of the spherical aberration on the intensity distribution at the paraxial focal plane becomes large with the increase of D, as can be seen in Fig. 6(b). Meanwhile, even under the influence of the spherical aberration, the intensity distribution at the best focal plane is found to be almost identical to that calculated with the aberration-free system.

Fig. 6 Calculated intensity distribution of the focused light spot on the SPD plane. (a) intensity distributions in 3-D plots (D = 2 mm); (b) intensity distributions in 3-D plots (D = 14 mm); (c) cross sections of the intensity distributions.

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The variation of the focused light spot diameter (1/e²) on the SPD plane with the different aperture diameter D is then calculated from the acquired intensity distributions as shown in Fig. 7. As can be seen in the figure, in the case of the aberration-free system, the focused light spot diameter continued to decrease as the increase of D. On the other hand, in the case of the system with the spherical aberration, the focused light spot diameter at paraxial focus is found to increase when D exceeds 11 mm. Meanwhile, even under the influence of the spherical aberration, the focused light spot diameter is found to be almost identical to that of the aberration-free system until the SPD plane is located at the best focal plane.

Fig. 7 Variation of the focused light spot diameter on the SPD plane.

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By using the calculated I₁(u,v) at each measurement laser beam diameter D, the optical angle sensor output is then simulated. Figure 8 shows a schematic of a calculation procedure for the simulation. At first, total power of the focused light spot on the whole SPD plane is calculated. After that, the displacement of the focused light spot Δu_i corresponding to the reflector tilt angle Δθ_Yi is calculated. The sum of the total power of the focused light spot on the SPD active area is then be able to be calculated. As a result of a series of the calculations, the angle sensor output u_{SPD_out} at each reflector tilt angle Δθ_Yi can be acquired.

Fig. 8 Calculation procedure for the simulation of the optical angle sensor output.

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Figures 9(a) and 9(b) show the results of the calculations with the measurement laser beam diameter D of 2 mm and 14 mm, respectively. In the figure, variations of the sensor outputs with the tilt motion of the reflector Δθ (u_{SPD_out}-Δθ curves) are plotted. Variations of the sensor sensitivities as a function of Δθ, which is calculated from the data in the u_{SPD_out}-Δθ curves ranging from Δθ of −5 arc-seconds to 5 arc-seconds, are also calculated, and are plotted in Fig. 9(c). When D is set to be 2 mm, as can be seen in Fig. 9(a), u_{SPD_out}-Δθ curves are almost the same regardless of the existence of the spherical aberration. On the other hand, when D is set to be 14 mm, the sensor sensitivity is found to be dramatically degraded at the paraxial focal plane. It should be noted that the sensor sensitivity is clearly degraded at the best focal plane under the influence of the spherical aberration, even though the laser beam diameter is almost the same as that of the aberration-free system, as shown in Fig. 7. From these results, it is revealed that slight disturbance in the intensity distribution results in the degradation of the sensor sensitivity.

Fig. 9 Simulated sensor outputs. (a) u_{SPD_out}-Δθ curves (D = 2 mm); (b) u_{SPD_out}-Δθ curves (D = 14 mm); (c) variation of the sensitivity as a function of D.

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4. Experiments and results

4.1 Experiments on an optical bench

To verify the feasibility of the computer simulation described in the previous section, experiments were carried out on an optical bench. At first, the influence of the spherical aberration on the focused light spot diameter was investigated. Figure 10 shows a schematic of the optical setup prepared on the optical bench. A laser beam with a wavelength of 685 nm emitted from the laser diode (LD) was collimated by using an aspherical lens. The collimated laser beam was then expanded by using a beam expander, and the expanded laser beam was made to pass through a diameter-variable iris. The purpose of employing the beam expander is not only to expand the laser beam diameter but also to make the intensity distribution of the laser beam passing through the iris as flat as possible. The laser beam passed through the iris with an aperture diameter of D was then made to go through a plano-convex collimator objective with a focal length f of 100 mm, and the focused light spot was monitored by a commercial beam profiler (BeamScan XYS/LL/5μm, Photon Inc.). At the beginning of the experiment, the position of the beam profiler along the optical axis was adjusted with a narrow laser beam with a diameter D of 2 mm, and was kept stationary during the experiment; namely, the beam profiler was placed at the position almost identical to the paraxial focus. The aperture diameter of the iris D was increased from 2 mm to 14 mm, and the variation of the 1/e² diameter of the focused light spot was measured. Figure 11 shows the results. In the figure, the focused light spot diameter calculated in the simulation described in the previous section is also plotted. A good agreement can be found between the simulation and the experiment.

Fig. 10 Optical setup for measurement of the focused light spot.

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Fig. 11 Measured focused light spot diameter.

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Following the experiment described above, an autocollimation unit in the optical angle sensor was also constructed on the optical bench to investigate the influence of the spherical aberration on the sensor sensitivity. Figure 12(a) shows the developed optical configuration. The reflective mirror in Fig. 10 was mounted on a PZT tilt stage, which was calibrated in advance of the experiments by using a commercial autocollimator (Möller-Wedel Optical GmbH, Elcomat 3000). In addition, the beam profiler in Fig. 10 was replaced with the autocollimation unit, which consists of a beam splitter and two SPDs; SPD1 for measurement, and SPD2 for reference. Both the SPD1 and SPD2 were placed on the focal plane of the collimator objective. A positioning of the SPD1 on the focal plane was carried out in such a way that the center of the focused light spot was made to coincide with the edge of the active area on the SPD1 plane. Meanwhile, the active area on the SPD2 plane was aligned to capture the whole energy of the focused light spot. It should be noted that the alignments of the SPDs were carried out by setting the iris aperture diameter D to be 2 mm, at the beginning of the following experiments; namely, the active area on the SPD1 plane was located at the position almost identical to the paraxial focus. As the SPD, one of the active areas on a dual-element Si PIN photodiode (Hamamatsu S4204) was employed (Fig. 12(b)).

Fig. 12 Setup for sensitivity evaluation. (a) a schematic of the optical setup; (b) a photograph of the photosensitive plane of the photodiode.

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By using the optical setup described above, experiments were carried out to investigate the sensitivity of the tilt motion measurement. In the experiments, a continuous sinusoidal tilt motion was applied to the reflective mirror via the PZT tilt stage. Photocurrent outputs from the SPD1 and SPD2 were converted into voltage outputs by trans-impedance amplifiers, and were recorded simultaneously by a digital oscilloscope. Figure 13(a) shows a part of the results. In the figure, the u_{SPD_out}-Δθ curves simulated in the previous section are also plotted. A variation of the sensor sensitivity, which is calculated from the data in the measured u_{SPD_out}-Δθ curves ranging from Δθ of −5 arc-seconds to 5 arc-seconds, is also calculated as shown in Fig. 13(b). As can be seen in the figure, the increase of the measurement laser beam diameter D was found to be effective in improving the sensor sensitivity until D exceeded 10 mm. However, on the other hand, further larger D did not contribute to the sensitivity improvement, since the influence of the spherical aberration of the collimator objective became significant, as predicted in the simulation. Although a slight difference can be found between the simulation results and experimental results, the tendency of the variation of the measured sensor sensitivity is quite similar with that acquired in the simulation. These experimental results confirm the relevance of the simulation results described in the previous section. One of the possible reasons of the slight difference between the experimental results and simulation results is the influence of the non-uniform sensitivity distribution of the SPD across its active area. According to the reference [23], a photodiode tends to have a transition region with the size of at least several-ten μm at its edge of the active area. The sensitivity distribution of the SPD active area at its edge is one of the most important factors in terms of the angle sensor sensitivity, and further investigation on this issue will be carried out as future work.

Fig. 13 Variation of the sensor output with the tilt angle applied to the reflective mirror. (a) u_{SPD_out}-Δθ curves acquired in the experiments; (b) sensor sensitivity.

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4.2 Experiments with a prototype angle sensor designed in a compact size

The results from the experiments on the optical bench have confirmed the feasibility of the concept for the optical angle sensor. As a next step of this study, an optical sensor head for one-dimensional prototype angle sensor was designed. The optical angle sensor will mainly be employed to evaluate small tilt angles (on the order of several arc-seconds) of precision instruments such as a precision linear stage and so on in a factory line, in which a compactness of the sensor head is of great importance. Therefore, attentions were paid not only to achieve high measurement resolution up to 0.001 arc-second, but also to design the optical sensor head in a compact size.

Figure 14(a) shows a schematic of the prototype optical sensor head. The sensor head was designed in a compact size of 100 mm × 150 mm. As a light source for the optical sensor head, the laser diode (LD) with the wavelength of 685 nm was employed again. In the design, the beam expander could not be employed due to the restriction of the available space. Meanwhile, regarding the simulation and experimental results described in the previous sections, the maximum diameter D of the measurement laser beam was 5 mm so that the benefit of larger measurement laser beam diameter D could be utilized. A collimating lens, which was provided by an optics manufacturing company, was selected for getting a measurement laser beam with a smooth intensity distribution profile with reduced interference and speckle noises. The 5 mm maximum diameter of the measurement laser beam was determined by the collimating lens. For this reason, investigation of the effect of using a beam diameter on the angular resolution was limited within 5 mm in the following experiment. It should be noted that it is desired to make the investigation also for a beam diameter beyond 5 mm for the consistency with the previous section and this will be carried out as a future work. An optical isolator consisting of a combination of a polarized beam splitter (PBS) and a quarter wave plate (QWP) was employed in the optical path of the sensor head to stabilize the laser power, which is one of the important factors in terms of the principle of the optical angle sensor. As a collimator objective, an achromatic doublet lens, which is expected to reduce the influence of the spherical aberration, was employed. Regarding the condition estimated in the simulation in the previous section, the focal length of the collimator objective was set to be 100 mm. Since the purpose of the development of the prototype optical sensor head was to verify that the measurement resolution of 0.001 arc-second could be achieved even by the sensor head designed in a compact size, only a single θ-component was included in the design. It should be noted that the optical sensor head still has enough space to include additional autocollimation units to increase the degree-of-freedom of the tilt angle measurement. In the setup, the same photodiodes employed in the experiments carried out on the optical bench were utilized, as well as the trans-impedance amplifier and the digital oscilloscope. A photograph of the experimental setup with the developed optical sensor head is shown in Fig. 14(b). A flat mirror was employed as a reflector, and was mounted on the PZT tilt stage. A whole the optical setup was built on a vibration isolation table.

Fig. 14 The optical sensor head designed in a compact size of 100 mm × 150 mm for the optical angle sensor. (a) a schematic of the optical configuration; (b) a photograph of the optical setup.

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By using the developed optical setup, some experiments were carried out. At first, a noise component of each SPD output through the trans-impedance amplifier, which is one of the most important parameters determining the measurement resolution of the developed angle sensor, was investigated. At first, the center of the focused light spot was aligned to be on the edge of the active area on the SPD1 plane by using the PZT tilt stage. The tilt stage was then kept stationary, and the outputs from the SPD1 and SPD2 were captured. Figure 15 shows a typical waveform of the SPD output. A standard deviation (σ) of the voltage outputs from both the SPD1 and SPD2 were confirmed to be approximately 0.3 mV, corresponding to the fluctuation of the sensor output of approximately 0.00323%.

Fig. 15 A typical waveform of the SPD output through the trans-impedance amplifier.

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The sensitivity of the angle sensor output was then evaluated. A continuous sinusoidal tilt motion about the Y-axis was applied to the reflective mirror, and a variation of the sensor output was recorded. Figures 16(a) and 16(b) show the u_{SPD_out}-Δθ curves from the computer simulation in the previous section and the experiments, respectively. In the experiments, the measurement laser beam diameter D was adjusted by inserting an aperture between the collimating lens and the polarized beam splitter as shown in Fig. 14(a). After that, the sensor sensitivity was calculated by using the data in the measured u_{SPD_out}-Δθ curves ranging from Δθ of −5 arc-seconds to 5 arc-seconds. As expected, the sensitivity of the prototype angle sensor was also improved in proportion to the measurement laser beam diameter D. In addition, a fairly good correlation can be found between the results of the computer simulation and experiments. When D was set to be 5 mm, the sensor sensitivity was 4.57%/arc-second. Regarding the fluctuation of the sensor output (0.00323%), the measurement resolution of approximately 0.0007 arc-second is expected to be achieved by the developed prototype sensor.

Fig. 16 Variation of the sensor sensitivity with respect to the measurement laser beam diameter D. (a) u_{SPD_out}-Δθ curves acquired by the computer simulation; (b) u_{SPD_out}-Δθ curves acquired by the experiments; (c) sensor sensitivity.

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To further verify the measurement resolution of the prototype angle sensor, dynamic tilt motion measurement was carried out by using the developed prototype sensor. Figure 17 shows the variation of the sensor output when a continuous sinusoidal tilt motion with amplitude and the frequency of 0.001 arc-second and 1 Hz, respectively, was applied to the reflective mirror by the PZT tilt stage. In the experiment, the measurement laser beam diameter D was set to be 5 mm. As can be seen in Fig. 17, the sinusoidal sensor output due to the mirror tilt motion was clearly observed. From these results, it can be concluded that the prototype angle sensor has achieved a measurement resolution of better than 0.001 arc-second with the sampling frequency of 1 kHz, while its optical sensor head is in a compact size of 100 mm × 150 mm. It should be noted that further better measurement resolution is expected to be achieved by employing an aspherical lens as the collimator objective to utilize the benefit from the larger measurement laser beam diameter D in the optimized configuration of the optical sensor head, as well as the employment of the laser diode with a shorter wavelength, applications of which will be carried out as future work.

Fig. 17 Measurement of an angular motion with the amplitude of 0.001 arc-second (measurement laser beam diameter D was set to be 5 mm).

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5. Conclusion

For measurement of tilt motions of precision instruments such as a precision linear stage, typical tilt angle of which is on the order of several arc-seconds, an ultra-sensitive optical angle sensor based on laser autocollimation has been developed. Since the optical angle sensor is expected to be used in a factory line, where compactness of a sensor head is of great importance, its optical sensor head is required to be designed in a compact size. In this paper, computer simulation based on wave optics has been carried out to investigate the influence of the spherical aberration of the collimator objective on the sensitivity of the optical angle sensor. An optical model with a single plano-convex lens has been established for simulating an autocollimation unit being employed in the optical sensor head to detect tilt motions of the reflector in a high sensitivity, and the influence of the spherical aberration on the angle sensor sensitivity has been evaluated. It has been verified from the simulation results that the increase of the measurement laser beam diameter D is effective in improving the sensor sensitivity although the influence of the spherical aberration becomes significant when D exceeds 10 mm. Experiments have also been carried out on an optical bench to verify the feasibility of the computer simulation. The measured focused light spot diameter has shown a good agreement with the simulation results, as well as the sensor sensitivity. Furthermore, a prototype optical sensor head having SPDs has been designed in a compact size of 100 mm × 150 mm, and has been developed to demonstrate the measurement resolution of better than 0.001 arc-second. Experimental results with the prototype optical sensor head with a maximum measurement laser beam diameter D of 5 mm, which was determined by the collimating lens, have revealed that the increase of D is effective in improving angle sensor sensitivity as predicted in the computer simulation. Furthermore, a tilt motion of the stage system with amplitude of ± 0.001 arc-second has successfully been distinguished with the sampling frequency of 1 kHz when D was taken as the maximum value of 5 mm. A design optimization of the optical sensor head with a larger beam diameter will be carried out to investigate the effect of using a beam diameter beyond 5 mm on the measurement resolution and to search the possibility of further improvement of the measurement resolution. Application of the sensor to the measurement of precision instruments such as a precision linear slide will also be carried out as future work.

Acknowledgments

This research was supported by the Japan Society for the Promotion of Science (JSPS), Mazak Foundation and Mitutoyo Science Foundation.

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Parameter	Value	Unit
Wavelength (λ)	685	[nm]
Focal length of the collimator objective (f)	100	[mm]
Refractive index of the collimator objective (n)	1.517	-
Aperture diameter on the pupil plane (,corresponding to the measurement laser beam diameter) (D)	1 to 14	[mm]

Ultra-sensitive angle sensor based on laser autocollimation for measurement of stage tilt motions

Abstract

1. Introduction

2. Principle of the ultra-sensitive optical angle sensor

3. Simulation on the angle sensor sensitivity

4. Experiments and results

4.1 Experiments on an optical bench

4.2 Experiments with a prototype angle sensor designed in a compact size

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (17)

Tables (1)

Equations (12)

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