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This paper reports an innovative heterodyne Wollaston laser encoder with common-optical-path (COP) configuration for the measurement of in-plane displacement. The proposed technique combines the design concepts underlying heterodyne interferometry, grating interferometry, and COP configuration. According to the measurement principle, displacement information pertaining to a moving grating can be obtained from variations in the optical phase induced by the grating. Several experiments were performed to evaluate the feasibility and performance of the proposed heterodyne Wollaston laser encoder. Experiment results demonstrate that the proposed method is capable of sensing in-plane displacement to a resolution of 2 nm while maintaining the stability of the system against environmental disturbances.
© 2016 Optical Society of America
1. Introduction
Interferometers are widely used in the semiconductor industry, precision manufacturing, metrology, biomedical engineering, and photolithography. These devices enable non-contact detection, high resolution, high responsivity, and long range measurement [1–6]. Various types of interferometer have been designed to deal with specific fields of application or objectives such as the measurement of displacement, velocity, refractive index, wafer warpage, or geometric errors. The importance of measuring displacement has led numerous academic institutions and firms to seek advancements in this technology with the aim of enhancing accuracy and reliability [7,8].
The scheme used for optical detection determines the classification of interferometers as homodyne [9–11] or heterodyne [12–18]. Heterodyne detection interferometry has proven superior to homodyne detection with regard the problem of high-frequency noise, which can seriously hinder efforts to achieve sub-nanometer resolution. The measurement principles used in heterodyne interferometry depend on a light source with two different optical frequencies. Many heterodyne modulation methods, such as the moving grating method, rotating plate method, acousto-optic modulator, electro-optic modulator, and Zeeman laser, were developed for the generation of a heterodyne light source [19–22]. Each of these methods comes with its own merits and limitations. Electro-optic modulators simplify signal processing and enable direct operations and several heterodyne interferometers have been developed using this technology [19,20, 23–25]. However, the problem of measurement accuracy and reliability caused by instabilities in the wavelength of light remains an issue.
Grating interferometers, also known as laser encoders, were developed to overcome problems associated with instability in the wavelengths of light, by decoupling the phase variation of the laser encoder from the wavelength of the light source. Numerous studies have used the measurement principle of laser encoders to measure displacement more precisely [9–13, 22–25]. However, most of the optical arrangements used in laser encoders belong to a non-common-optical-path design configuration, thereby rendering measurement results susceptible to environmental disturbances and necessitating the use of methods for compensation. In our previous study [25], we proposed a heterodyne grating-shearing interferometer based on the quasi-common-optical-path (QCOP) configuration for the measurement of displacement with six degrees of freedom (DOF). According to the measurement principle of the QCOP configuration, the optical paths of the reference and measurement beams are nearly equivalent. This makes it possible to compensate for environmental disturbances using interference signals and thereby achieve system stability without the need for further compensation. Nonetheless, this requires a specific semicircular half-wave plate (HWP) for rotating the polarization states of the half heterodyne light beam. The manufacture of this specific HWP requires a specific cutting angle which means this element is difficult to acquire. Moreover, errors associated with the specific cutting angle during the manufacture, incorrect azimuth angle, and material quality of the semi-circular HWP will lead to non-linear measurement error. The QCOP configuration also requires a focusing lens to partially overlap the zeroth and first order diffraction beams in order to form an interference pattern. Therefore, a defocus problem might be induced if the grating element undergoes out-of-plane displacement. These problems limit the development and applications of the QCOP configuration.
In this paper, we propose an innovative heterodyne Wollaston laser encoder (HWLE) using a Wollaston prism (WP) to achieve a common-optical-path (COP) configuration to overcome the problems mentioned above. The construction of the measurement technique involves heterodyne interferometry, Wollaston interferometry, grating interferometry, and COP configuration. The proposed HWLE does not require any optical elements, such as a HWP, for the rotation of the polarization states of the heterodyne light beam. The problem of defocus does not exist in this configuration. Moreover, the COP configuration requires fewer optical elements than the QCOP, which allows easier set-up and reduces error due to misalignment. These show the superiority and potential of the proposed COP configuration. The measurement principle underlying the proposed HWLE is detailed in the following. Experiment results demonstrate the feasibility and performance of the proposed system.
2. Measurement principle
2.1. Design concept of common-optical-path configuration
For convenience, the z-axis was set to follow the direction of light propagation, and the x-axis was set as the horizontal plane. As shown in Fig. 1(a), a light beam normally enters the optical element of a Wollaston prism (WP), which consists of two calcite right-angled prisms optically cemented together at the hypotenuse. The optical axes of each prism are simultaneously perpendicular to one another and to the propagating beam. It is worth mentioning here that WP elements are easily obtained and the extinction ratio of a typical WP is sufficiently high (>20000:1) to avoid the problem of non-linear measurement error. The incoming light beam traversing the first prism is not split, but the ordinary ray (o-ray) is retarded with respect to the extraordinary ray (e-ray). Upon entering the second section, the o-ray (s-polarized beam) is bent toward the interface normal while the e-ray (p-polarized beam) is bent away from the interface normal. This means that the s- and p-polarized components of the incoming beam diffract in opposite directions. It is worth mentioning that the o-ray and e-ray are symmetric to each other. Moreover, prisms with a separation angle (δ) from about 5° to 45° between the exiting beams can also be manufactured and used according to the wedge angles of the right angle prism. According to the working principle of WP, when the light beam passes through the WP, the optical phase variations (ϕp and ϕs) of the s- and p-polarized states can be expressed as
where no and ne are the ordinary refractive index and extraordinary refractive index, 𝜃 and λ represents the incident angle of the beam at the hypotenuse and the wavelength of the light beam, lo and le represent the optical path distances of the o-ray and e-ray in the second right angle prism, which is given as follows:where 𝜃o and 𝜃e are the refraction angles of the o-ray and e-ray, respectively. As determined using Eq. (1) and Eq. (2), the optical phase variations of the s- and p-polarized states are functions of x, belonging to the spatial distribution function. Moreover, the Jones matrix of the WP can be written as follows:
Fig. 1 (a) Light beam passing through WP optical element; (b) Expanded light beam passing through WP optical element and corresponding grating.
As shown in Fig. 1(b), a diffraction grating is placed behind the WP. When an expanded light beam passes through the WP and the diffraction grating, the s- and p-polarized beams diffract. By selecting a suitable separation angle for the WP and pitch size for the diffraction grating, the corresponding diffracted beams partially overlap and interfere. For example, the zeroth order p-polarized diffracted beam (P0) overlaps with the positive first order s-polarized diffracted beam (S+1), whereas the zeroth order s-polarized diffracted beam (S0) overlaps with the negative first order p-polarized diffracted beam (P-1). When the grating undergoes a displacement (dx) along the x direction, the optical phase (ϕm) of the mth order diffracted beam varies as follows:
where p is the grating pitch. Therefore, when the grating moves along the x direction, changes in the interference phase can be observed from the overlapping area. In addition, the optical paths of the two orthogonally polarized beams in the overlapping areas are almost equivalent, which means they are similarly affected by surrounding disturbances. Nonetheless, compensation can be applied to deal with the interference signal. This design is regarded as a COP configuration.2.2. Design of heterodyne Wollaston laser encoder
Figure 2 illustrates the optical configuration of the proposed HWLE. As shown in the figure, the light beam emitted from the laser source is linearly polarized at 45° with respect to the x-axis, after passing through the polarizer. A sawtooth signal with half-wave-voltage is applied to the electro-optic modulator (EOM) to generate a heterodyne light source. According to Su’s principle [19], the complex amplitude of the heterodyne light beam can be written as follows:
where ∆ω represents the modulation frequency of the light source. A non-polarizing beam-splitter (BS) is used for the separation of the original heterodyne light beam into two beams moving along two different optical paths. The reflection beam is used to produce a reference heterodyne signal while the transmission beam is used to establish interference. Then, a beam expander (BE) is used to enlarge the original heterodyne light beam. The expanded light beam enters the WP normally. In our Then, a beam expander (BE) is used to enlarge the original heterodyne light beam. The expanded light beam enters the WP normally. In our case, the separation angle of WP was set to approximately 20°. According to the Jones calculation, the electric fields of the s- and p-polarized beams are given as follows:Moreover, the two exiting beams pass through the grating before being diffracted and partially overlapped. According to the design concept mentioned in the previous section and the Jones calculation, the electric fields of each diffracted s- and p-polarized beams can be described as follows:where ϕ0 = 0 and ϕ ± 1 = ± 2πdx/p. The insert in Fig. 2 shows the two overlapping areas of the diffracted beams (O1 and O2) with interference fringes. The electric fields in O1 and O2 are written as follows:Two analyzers (AN1 and AN2) with transmittance axes at 45° are then placed in front of detectors PD1 and PD2 to obtain interference beating signals (I1 and I2), which can be expressed as follows:As can be seen in Eq. (9), the interference signals (I1 and I2) obtained by PD1 and PD2 are the same, even when the grating moves, which means that we can choose either I1 or I2 as the measurement signal. Regardless of the distance between the grating and detector, the interference signals of the overlapping areas remain unaffected by other diffraction beams. Meanwhile, the interference signal (Ir) detected by PDr can be regarded as a reference signal, given as follows:where ϕr refers to the initial phase of beam r. The reference signal (Ir) and the measurement signal (I1 or I2) are sent into a lock-in amplifier for phase demodulation and calculation. For example, the phase difference (Φ) between Ir and I1 is described asIn the following, we seek to verify our assumption that ϕp, ϕs , and ϕr are constants and can thus be ignored. Clearly, the calculation of in-plane displacement (dx) can be based on the measurements of phase difference variation and the known grating pitch (p). The relationship among the displacement, phase difference variation, and the grating pitch can be written as follows:Fig. 2 Schematic illustration showing optical configuration of heterodyne Wollaston laser encoder, comprising a beam expander (BE), electro-optic modulator (EOM), polarizer (P), analyzers (AN1, AN2), detectors (PD1, PD2, PDr), Wollaston prism (WP), grating, and iris.
Clearly, the in-plane displacement information can be obtained using the proposed method. As previously mentioned, the design concept of COP configuration relies on the fact that the optical paths of diffracted s- and p-polarized beams are nearly the same and that they both suffer from similar environment disturbances, which means that disturbances can be compensated for in the interference signal.
3. System performance test
3.1. Experiment setup
Several experiments were conducted to determine the feasibility and performance of the proposed method. The experiment setup is presented in Fig. 3. A heterodyne light source from a He–Ne (λ = 632.8 nm) laser was modulated using an electro-optic modulator (Newport co. model: 4002) for amplitude modulation. The difference in frequency between the p- and s-polarizations of the heterodyne light source was set at 16 kHz and a beam expander (Thorlabs, Inc. model: BW10M) was used to enlarge the heterodyne light beam. A diffraction grating (Edmond, 500 grooves/mm) was mounted on a commercial 6-DOF positioning stage (Physik Instrumente, model: PI-P-562.6CD) in conjunction with internal capacitive sensors. A Wollaston prism (Thorlabs, Inc. model: WP10) with a beam separation angle of 20° was selected to direct the heterodyne beam into the grating. The selection of a suitable grating pitch enables the corresponding diffraction orders to partially overlap, as previously mentioned with regard to the measurement principle. It is worth mentioning that the symmetry in the diffraction order pattern greatly facilitates set-up and alignment. The interference signals are measured after passing through the polarizers and photo-detectors (Thorlabs, Inc. model: PDA-36 EC). A software lock-in amplifier was programed using graphical language (National Instruments, LabVIEW) for demodulation and the calculation of variations in phase difference between interference signals. In-plane displacement (dx) can then be determined according to variations in measured phase difference (Φ).
3.2. Displacement tests: short- and long-range
To evaluate the efficacy of the HWLE in the measurement of short- and long-range displacements, the positioning stage was operated in a closed-loop function to enable forward and backward motion with a traveling range from 100 μm to 1 μm. Three waveform motions were executed, including sinusoidal, triangular, and trapezoidal waveforms. The resulting curves obtained using this method and the internal capacitive sensor are presented in Fig. 4. The sampling rate was set at 500 kHz and the calculation rate (fCR) of the software LIA was 125 Hz. To facilitate observation, the results are plotted with a time-delay to differentiate between the results obtained using the two measurement methods. As shown in Fig. 4, the measurement curves show similar displacements and motion behaviors, regardless of whether the proposed method (dotted line) or the internal capacitive sensor (solid line) were used. The measurement results obtained using the proposed method present the same degree of linearity as do those obtained using the internal capacitive sensor. This is a clear demonstration that the HWLE has the capacity to measure short- and long-range displacements, even in commercial applications.
To demonstrate the efficacy of the proposed method in a real-world testing situation, we also performed an experiment using random signals, the results of which are presented in Fig. 5. As shown in the figure, the experiment results show similar displacement values, regardless of whether a periodic signal or a random signal was used. This provides solid proof that the HWLE is able to measure in-plane displacement at resolutions comparable to those of commercial capacitive sensors in real-world testing situations.
Furthermore, according to the explanation in related studies [6,9,25], measurement resolution is defined as the minimum detectable displacement that can be achieved. To test the measurement of minimum detectable displacement of our method, the positioning stage was asked to perform step motion with a traveling range of 2 nm. The measured curves obtained using the two methods are plotted in Fig. 6. Again, the measurement result obtained using the proposed HWLE present high degree of linearity as the one obtained using the capacitive sensor. As shown in figure, the displacement resolution of our method can reach 2 nm. This clearly demonstrates that this HWLE has the ability and potential to provide nano-metric resolution.
3.3. Repeatability test
Repeatability is a major performance index applicable to every measurement technique. The repeatability of a system is inversely proportional to the deviation between the starting position and the final position. To test the repeatability of the HWLE, the positioning stage was moved over the distances of 50 nm and 10 nm in steps of 10 nm and 2 nm respectively, using the proposed laser encoder as well as an internal capacitive sensor using to measure the step movements. The sampling rate was set at 200 kHz and the calculation rate (fCR) of the software LIA was 50 Hz which enable a maximum detection frequency of 1 MHz. The experiment results obtained through the two methods are presented in Fig. 7, wherein the deviation between the starting position and the final position is approximately 1 nm. The results obtained using the HWLE are in good agreement with those obtained using the internal capacitive sensor. This is a clear demonstration of the high degree of repeatability of which the proposed HWLE is capable.
Moreover, the measurement resolution of our method can be determined by checking the minimum detectable displacement that can be achieved [6,9,25]. As can be seen in Fig. 7, the results of repeatability test also demonstrate that the minimum detectable displacement (measurement resolution) of our method can reach 2 nm. After calculating the experiment result, the standard deviation of the measurement error of the HWLE was approximately 1.34 nm while the one of the internal capacitive sensor was approximately 1.70 nm. This strongly indicates the proposed HWLE has high measurement performance.
3.4. Limitations: speed of measurement
In the current system, the measurement speed was limited mainly by the software lock-in amplifier (LIA) because the optical modulation frequency was higher than the calculation rate (fCR) of the software LIA. In theory, the relationships among the phase difference variation rate (dΦ/dt), measurement sensitivity, and measurement speed (d(dx)/dt) can be written as (dΦ/dt = s × d(dx)/dt). To prevent a loss of data, the phase difference variation rate should be lower than the calculation rate (fCR) of the software LIA; i.e., dΦ/dt = s × d(dx)/dt < π × fCR. Clearly, the calculation rate (fCR) of the software LIA limits the speed of measurement. In this section, the fCR in the measurement speed test was approximately 1176 Hz (850 μs). With measurement sensitivity s = 0.18°/nm, the theoretical measurement speed would be approximately 1176 μm/s.
To determine the maximum speed of measurement of which the HWLE is capable, a precision linear stepper (model: XS-50; Measure control, Inc.) was used to move the stage over a traveling range of 20 mm at speeds of 200 μm/s, 400 μm/s, 600 μm/s, 1100 μm/s, and 1200 μm/s. The stepper was integrated with a linear encoder sensor and operated in a closed-loop mode to enable the simultaneous measurement of the stepper movements for comparison. Figure 8 presents experiment results obtained using the HWLE and the linear encoder. The curves show similar displacements and behaviors except the measurement speed of 1200 μm/s. As shown in the left insert in Fig. 8, the measurement values obtained using the proposed method (green pentagon) are below the values acquired using the linear encoder (solid black line), due to the problem of phase unwrapping. In fact, when the rate of wrapped phase is higher than the rate at which data is output from the software LIA, data crucial to the measurement results may be missed. Thus, the maximum reliable speed of measurement is approximately 1100 μm/s. This experiment result confirms the theoretical estimation of the measurement speed mentioned above. We are currently in the process of developing an FPGA-based lock-in amplifier to enable measurement at higher speeds (e.g., 50 mm/s). Nonetheless, it is important to stress that the speeds achieved in the current experiments are compatible with the results of existing large-scale near-field probing microscopy [2].
3.5. Stability test
To demonstrate the ability of the proposed COP configuration in resisting environmental disturbance without compensation, we implemented a heterodyne grating interferometer (HGI) using a non-common-optical-path (NCOP) configuration with which to perform system stability tests for comparison [13]. The sampling rate was set at 200 kHz and the calculation rate (fCR) of the software LIA was 50 Hz. We sought to detect the variations in phase difference when the positioning stage was held stationary for five minutes. Figure 9 shows the measured variation of the phase difference over a period of five minutes. Because displacement measurement deals with dynamic measurement, system stability should be considered as a whole. As can be seen from the measured curves, the system stability of the proposed method was approximately 10 nm while the one of the NCOP configuration was approximately 50 nm, without any compensation. This result clearly illustrates how much less sensitive to the surrounding disturbance is the proposed COP configuration as compared to the NCOP configuration. Moreover, as can be seen from the two local inserts, according to the experiment results, the amplitude of the standard deviation of the noise of the HWLE was approximately 0.3 nm while the one of the HGI was approximately 1.4 nm.
Based on our previous experience and the findings obtained in related studies [20,26], the measurement result of system stability might be composed of both low-frequency drift and high-frequency components. In general, low-frequency drift is commonly the result of environmental disturbances or erroneous signal for heterodyne modulation. Environmental disturbance, such as thermal drift and mechanical vibration, could be reduced through the use of isolation cage, isolation system, or smaller pitch size [13,23]. There are two situations which commonly cause low-frequency drift as a result of incorrect modulation sawtooth signal. One situation involves an incorrect waveform of the modulation sawtooth signal, the other is the operating voltage applied to the EOM is not equal to the half-wave-voltage. When one of the situations occurs, the measurement result will be affected. This problem can be efficiently eliminated through careful adjustment of the external modulation sawtooth signal. High-frequency noise generated by system components, such as the laser source, electronic signals, or photodetectors is difficult to avoid; however, our current results demonstrate the high stability of the HWLE system even without compensation.
4. Discussion
4.1. Measurement range, sensitivity, and resolution
The range of displacement measurement using the proposed HWLE depends on the width of the diffraction grating used in the system. At present, the maximum measurement range is approximately 10 cm; however, a wider diffraction grating could be used to extend this. Nonetheless, the current measurement range is sufficiently large to allow for large-scale near-field probing microscopy [2]. According to the measurement principle in Eq. (12), the measurement sensitivity (s) of the proposed method can be written as follows:
In the current set-up, the diffraction grating has the pitch size of 2 μm. Following the substitution of the experimental parameters into Eq. (13), we obtained measurement sensitivity (s) = 0.18°/nm. Reducing the pitch size could further increase measurement sensitivity if required.
As mentioned previously with regard to displacement and repeatability tests, the resolution of the system can be determined by checking the minimum detectable displacement. In the current set-up, the grating pitch and minimum detectable phase resolution of the software LIA were approximately 2 μm and 0.01°. Thus, according to the measurement principle, the theoretical measurement resolution of the system would be 0.056 nm; however, the actual measurement results obtained in the displacement test and repeatability test revealed resolution of approximately 2 nm.
4.2. Periodic nonlinearity error analysis
The development of the HWLE was based on the technique of heterodyne interferometry; therefore, we paid particular attention to the error effects associated with interferometric technique. In form of heterodyne interferometry, measurement performance is influenced by non-linear periodic error resulting from the errors associated with frequency mixing, polarization mixing, and polarization-frequency mixing [17]. In fact, we analyzed non-linear periodic error for heterodyne grating-based interferometers in a previous study [23]. In this section, we follow a similar method of analysis in estimating the total nonlinear periodic error in the HWLE.
In the proposed system, most of the nonlinear error is due to misalignment, the extinction ratio, the WP, and the analyzer. In estimating the total error resulting from these sources, the expression of the interference amplitude in overlapping area (O1) in Fig. 2 can be written as
where α and δWP are the extinction ratio and error in the azimuth angle of the WP component, β and δAN represent the extinction ratio and error in the azimuth angle of the analyzer in front of the photo-detector, respectively. The intensity obtained by detectors can be written as follow:where Φ′ represents the deformed phase. The corresponding variation in the phase error (ΔΦ) of the HWLE can be obtained as follows:In the current experiment set-up, careful adjustments to minimize discrepancies in the azimuth angle of the WP and analyzer make it possible to control angle misalignment to within 5′ at the rotation mounts (Thorlabs, Inc. model: PRM1GL10). After substituting the related parameters into Eq. (16), we obtain non-linear periodic error of approximately ± 2.6 nm for every grating pitch. In fact, our experiments revealed the actual non-linear error at approximately ± 5 nm for every grating pitch. Based on our experience, we surmise that the difference between the estimations and results may be due to problems related to elliptical polarization from the light source or a lack of accuracy in the alignment of the experiment set-up.
4.3. Error resulting from misalignment of diffraction grating
Misalignment between the grating and the direction of light beam propagation is another key source of error that is difficult to avoid. Furthermore, it can seriously affect the measurement of displacement. Generally, three situations are likely to induce misalignment error: misalignment in the angles of pitch, yaw, and roll. As shown in Fig. 10, assembly error due to the angles of yaw, pitch, and roll can lead to the rotation of diffracted s- and p-polarized beams at the angles of θz, θy, and θx, respectively. These three situations are discussed below.
Case 1: Misalignment in the yaw angle (θz) can have measured effects on IP displacement in the x direction. According to the measurement principle, the amount of influence (Δde) induced by a misalignment in the yaw angle can be written as follows:
where ∆de and dx are the displacement error and real displacement, Ф is the change in phase, and pyaw and p are the grating pitch with or without misalignment error, respectively. If the misalignment in yaw angle were 0.1°, then the corresponding displacement error would be approximately 1.5 nm for 1 mm displacement.Case 2: Misalignment in the pitch angle (θy) can alter the pitch size of gratings in the x direction. As a result, the phase variation error caused by misalignment is coupled to the measurement of in-plane displacement. According to the measurement principle, the relationships among displacement, variations in phase difference, grating pitch, and measurement error can be written as follows:
where ∆de and dx are the displacement error and real displacement, Ф is the change in phase, and ppitch and p are the grating pitch with or without the misalignment error, respectively. If the misalignment in pitch angle were 0.1°, then the corresponding displacement error would be approximately 1.5 nm for 1 mm displacement.Case 3: Misalignment in the roll angle (θx) can alter the positions of the ± 1 order diffracted s- and p-polarized beams upward or downward along the y direction. However, this does not result in a change in the phase shift between the overlapping area of 0th and ± 1 order diffracted s- and p-polarized beams in the x direction. In other words, phase error caused by misalignment in the roll angle does not influence the measurement of displacement.
5. Conclusion
This paper reports an innovative heterodyne Wollaston laser encoder (HWLE) based on a common-optical-path (COP) configuration for the measurement of in-plane displacement. The design concept of the proposed HWLE relies on the fact that the optical paths of diffracted s- and p-polarized beams are nearly the same. This makes it possible to achieve measurement resolution at the nanometric level while maintaining high system stability. According to the working principles and the Jones calculation, displacement information related to a moving grating can be obtained by analyzing variations in optical phase induced by the grating. The theoretical resolution and sensitivity of the HWLE were estimated at 0.056 nm and 0.18°/nm, respectively.
Several experiments on the measurement of displacement as well as a systematic comparison with internal capacitive sensors and linear encoders demonstrate the feasibility and performance of the proposed HWLE. In experiments, we achieved resolution of 2 nm, repeatability of 1 nm, and speed of 1100 μm/s. These results demonstrate that the HWLE is capable of measuring in-plane displacement with a high degree of precision while maintaining outstanding system stability.
Acknowledgments
This study was supported by the Ministry of Science and Technology (MOST), Taiwan, under contract MOST 104-2221-E-011-116. The authors cordially thank Prof. J. Y. Lee (National Central University, Taiwan) and Mr. K. A. Huang (National Taiwan University of Science and Technology, Taiwan) for their assistance.
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