1. Introduction

In recent years, with the rapid development of semiconductor, advanced manufacturing and other scientific researches, there are great demands for displacement measurements applicable to millimeter range with sub-nanometer resolution [1–3]. For example, the width of the smallest structures of integrated circuits continuously decreases from 125nm to 45nm as the wafer size increases from 200 mm to 300mm, the manufacturing accuracy has reached to about 10 nm as the motion range of the stage of CNC machine tool is up to 100 mm and the calibration of scanning probe microscopes or transducers in the millimeter range with nanometer accuracy [4–7]. Typically, conventional nanometrology can be divided into two kinds of approaches. One is the non-optical measurement techniques including scanning tunneling microscope, atomic force microscope (AFM) and capacitive sensors, etc. Another is the optical measurement techniques including Michelson interferometer, heterodyne interferometer and X-ray interferometer, etc. Although the non-optical measurement techniques such as AFM can achieve sub-nanometer resolution, the measurement range is limited to micrometer level, and these instruments need to be traceable to the definition of the meter by using optical methods. Through subdivision and counting of interference fringes, both Michelson interferometer and heterodyne interferometer can realize large displacement measurement. However, the improvement of measurement accuracy is limited due to the sinusoidal error caused by lack of quadrature in Michelson interferometer or the first-order phase error caused by polarization nonorthogonality of the laser beams and misalignment in heterodyne interferometer [8–10].

In order to realize displacement measurement in millimeter range with sub-nanometer resolution, a laser synthetic wavelength interferometer (LSWI) is developed in this paper. The paper is structured as follows: section 2 provides the description of the LSWI design and its operation principle. In section 3 the signal processing of the LSWI is described. Section 4 describes displacement measurement experiments and comparison between the LSWI and an Agilent 5529A interferometer, while in section 5 the resolution and nonlinearity of the LSWI are discussed.

2. Measurement principle

The system of the LSWI is shown in Fig. 1 . Beams λ₁ and λ₂ are two orthogonal linearly polarized beams emitted by a dual wavelength He-Ne laser. The frequency difference between the two beams is approximately 1 GHz. The beam λ₁ is split into two beams with almost equal intensity by a beamsplitter (BS₁), one directed to the reference corner cube (M₁) and the other to a polarizing beamsplitter (PBS₁). The reflected beams are recombined in BS₁ and reflected by another polarizing beamsplitter (PBS₂), and then the interference signal is detected by a photodetector (PD₁). Similarly, the beam λ₂ is split into two beams with almost equal intensity by BS₁, one directed to M₁ and the other transmitted through PBS₁ to a moving corner cube (M₂). They are reflected back, get together in BS₁ and then pass through PBS₂. This interference signal is split by another beamspliter (BS₂) into two signals detected by PD₂ and PD₃, respectively. PD₂ and PD₃ are placed at an interval of quarter fringe in order to produce a cosine signal and a sine signal. The determination of the phase difference between signals of PD₁ and PD₂ and the bidirectional fringe counting with PD₂ and PD₃ are completed in the signal processing unit. When M₂ is moved a measured nanometer displacement Δl less than half a wavelength, then the signal processing unit sends a signal to the stage controller, so M₁ is moved a compensative displacement ΔL in order to keep the phase difference constant. Through theoretical derivation [11,12], the measured displacement Δl expressed by

 

Fig. 1 Schematic diagram of the laser synthetic wavelength interferometer system.

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This shows that a smaller displacement Δl to be measured can be realized by determining a larger displacement ΔL because the synthetic wavelength λ_S is much greater than the single wavelength λ₂. From Eq. (1), suppose the wavelength λ₂ = 632.8 nm and the synthetic wavelength λ_S = 280 mm, it is obvious that 1 nm measured displacement of M₂ corresponds to 0.44 mm displacement of M₁, and 0.01 nm measured displacement of M₂ to 4.4 µm displacement of M₁. That is, if we measure 0.44 mm or 4.4 µm displacement of M₁, we can achieve 1 nm or 0.01 nm measured displacement of M₂. This shows that the LSWI is expected to realize a capability of sub-nanometer resolution in displacement measurement.

Combining the integer fringe counting, a large displacement of M₂ is given by

Equation (2) shows that the LSWI can achieve large displacement measurement with nanometer accuracy.

3. Signal processing

According to the above description, the signal processing of LSWI includes interference signals processing, compensation of air refractive index and control of the movement of M₁.

3.1 Interference signals processing

As shown in Fig. 2 , the curves V(λ₁), V(λ₂)cosine and V(λ₂)sine represent the interference signals received by PD₁, PD₂ and PD₃, respectively. The curves V(λ₁) and V(λ₂)cosine are used to determine the phase difference between the interference signals λ₁ and λ₂, and the V(λ₂)cosine and V(λ₂)sine are used to count the integer fringe number of beam λ₂ and to judge the moving direction of M₂. The filling pulse number n₀ represents the original phase difference before M₂ starts moving. The dashed curve V'(λ₂)cosine signifies the signal when M₂ is moved a certain displacement corresponding to a fractional fringe, the filling pulse number n₁ signifies current phase difference. To keep the phase difference constant, M₁ is moved forward or backward according to the moving direction of M₂. Thus, n₁ approaches to n₀. In addition, when M₂ is moved a displacement corresponding to integer fringes, the integer fringe number N is calculated by the software fringe counting method [12]. The signal V(λ₁) is invariable during the moving of M₂, thus the signal V(λ₁) can be regarded as a reference signal in order to realize the combination of fraction and integer fringes.

 

Fig. 2 A diagram illustrating how to realize interference signals processing.

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3.2 Compensation of air refractive index

Taking into account air refractive index in Eq. (2), the displacement Δl is expressed by

Equation (3) signifies that in the high-precision displacement measurement, we must obtain the accurate air refractive index in real time. According to Birch and Downs research [13], supposing the changes of CO₂ content is negligible, the revised dispersion formula for standard condition air at 1 Pa and 15 °C is

For the moisture air containing a partial pressure f of water vapor, the dispersion formula is expressed by

In the above equations, the temperature T is expressed in degree Celsius, the atmosphere pressure P in Pa and partial pressure f in Pa.

Therefore, based on the above equations, we can calculate accurate air refractivity by measuring the parameters of pressure, temperature and humidity to amend the displacement measurement result.

3.3 Implementation of signal processing

The signal processing unit in the LSWI is shown in Fig. 3 . A digital signal processor (TMS320F28335 DSP, Texas Instruments Co., USA) is used to act for data sampling, processing and sending. The interference signals detected by PD₁ and PD₂ are filtered, amplified and converted to square waves to realize the determination of the phase difference. Interference signals of λ₂ detected by PD₂ and PD₃ are filtered and amplified for integer fringe counting. The real-time parameters of air condition are detected by a pressure sensor (MS5534C, Intersema Co., Switzerland), a humidity sensor (DB113-15, DABECO Co. China) and a temperature sensor (TSIC506F, ZMD Co. Germany). The fringe counting and real-time compensation of air refractive index was implemented in DSP. The phase difference determination and integer counting results are transmitted to a personal computer (PC) for combination of fraction and integer fringe counting. The PC sends control signals to the controller (C862, Physik Instrumente Co., Germany) of a stage to drive M₁ according to the phase difference variation.

 

Fig. 3 A schematic diagram of the signal processing of LSWI.

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4. Experiment and results

To verify its performance, the LSWI was developed as shown in Fig. 4 . The laser source was a dual-wavelength He-Ne laser (ML10, Renishaw Co., UK), which emits a pair of beams with the central wavelength of 632.99130 nm and the frequency difference of 1085 MHz. The synthetic wavelength λ_S is then 276.3064 mm. The M₁ displacement ΔL was provided by a linear stage (M-531.DD, Physik Instrumente Co., Germany) whose movement rang is 300 mm with 0.1 μm resolution. The measured displacement of M₂ was provided by a linear stage (M-521.DD, Physik Instrumente Co., Germany) whose movement rang is 200 mm with 0.1 μm resolution and a nano-positioning stage (P-752.1CD, Physik Instrument Co., Germany) whose movement rang is 15 μm with resolution of 0.1 nm. And a commercial heterodyne interferometer (5529A, Agilent Co., USA) with 10 nm resolution was used to test the displacement of the same stage for comparison.

 

Fig. 4 Photo of the experimental setup.

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As the experiment began, firstly, the M₁ was moved to find a position where the phase difference between the two interference signals λ₁ and λ₂ was set an original value of zero. Secondly, the M₂ was moved a measured displacement Δl, the integer fringe number N was calculated by the DSP. Thirdly, the M₁ was moved forward or backward according to the moving direction of M₂ to make the phase difference reach to the original value of zero again, the M₁ displacement ΔL was recorded. Substituting N and ΔL into the Eq. (2), the measured displacement Δl was obtained. The measurement results were compensated according Eqs. (4) - (6). Three experiments were carried out in millimeter range, micrometer range and nanometer range, respectively.

4.1 Displacement measurement experiment in millimeter range

In this experiment, measurements were performed by moving the M-521.DD stage for a distance of 150 mm divided into 30 steps. The step displacement was 5 mm. These displacements were measured simultaneously with the LSWI and the Agilent 5529A interferometer. The velocity of M₁ was set 1 mm/s, so the maximum measurement time was about 138.15 s when the phase difference approached to 360°, and the average time for each of the measurements was about 79.08 s. The experimental result is shown in Fig. 5 and summarized in Table 1 . The deviations are the differences between the actual displacements of the M-521.DD stage and the measurement results with the LSWI or with the Agilent 5529A interferometer. Linear fitting of the experimental data can obtain the linearity coefficients shown in Table 1. Figure 5 and Table 1 show that the results with the LSWI are consistent with those obtained from the Agilent interferometer. These results demonstrated that the LSWI can be applied to millimeter range displacement measurement.

 

Fig. 5 Comparison of displacement measurements in millimeter range.

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Table 1. Displacement measurement results in millimeter range

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4.2 Displacement measurement experiment in micrometer range

In this experiment, measurements were performed by moving the nano-positioning stage P-752.1CD for a distance of 14 µm divided into 28 steps. The step displacement was 0.5 µm. These displacements were measured simultaneously with the LSWI and the Agilent 5529A interferometer. The experimental result is shown in Fig. 6 and summarized in Table 2 . This shows that the results with the LSWI are consistent with those obtained from the Agilent interferometer. These results demonstrate that the LSWI can achieve nanometer level accuracy for displacement measurement.

 

Fig. 6 Comparison of displacement measurements in micrometer range.

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Table 2. Displacement measurement results in micrometer range

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4.3 Displacement measurement experiment in nanometer range

In this experiment, the M₂ was moved with step increment 10 nm in the range of 450 nm. Figure 7 shows the experimental result with the LSWI and the deviation compared with the P-752.1CD stage’s displacement. The experimental result indicates that the average error is 0.03 nm, the standard deviation is 1.4 nm and the linearity coefficient is 0.9999. This further demonstrates that the LSWI can achieve nanometer accuracy because the errors shown in Fig. 7 do not display obvious sinusoidal nonlinearity within a displacement range of half a wavelength.

 

Fig. 7 Displacement measurements result in nanometer range.

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5. Discussion

5.1 Resolution analysis

From Eq. (1), considering the limit of measurement accuracy influenced by sources of error, such as the laser source stability, the M-531.DD stage positioning accuracy and the variation of air refractive index, the uncertainty of the LSWI is expressed by

Here δλ₂ represents the uncertainty of single wavelength, δλ_S represents the uncertainty of synthetic wavelength, δΔL is the positioning accuracy of the stage and the δr/r takes into account the errors of Δl induced by the uncertainty of ratio n(λ_S)/ n(λ₂) of the air refractive induces at the two wavelengths λ_S and λ₂. The maximum displacement of ΔL is λ_S/2.

According to the parameters provided by the ML10 laser manual, the frequency f₁ = 473612829 MHz and f₂ = 473611744 MHz. The frequency stability is 0.05 ppm. So the variations of δλ₂, δλ_S can be calculated as

The positioning accuracy of the stage δΔL is determined mainly by the moving velocity of the stage and the frequency of fast clock shown in Fig. 2. The δΔL is expressed by

Here λ is the central wavelength of 632.99130 nm, ν is the moving velocity of the stage and f_clk is the frequency of fast clock.

Substituting ν = 1 mm/s and f_clk = 150 MHz that we used into Eq. (12), we can get δΔL = 2.91µm

Basing on the researches of Owens and Castell [14,15], the uncertainty of ratio n(λ_S)/ n(λ₂) can be expressed by

Here $a(\lambda )=\left(2371.34+\frac{683939.7{\lambda }^2}{130{\lambda }^2-1}+\frac{4547.3{\lambda }^2}{38.9{\lambda }^2-1}\right)\times {10}^{-8},$

In Eq. (13), the density factors D_S of dry air at standard conditions (0.03% CO₂) and D_W of water vapor are related to the partial pressures P_S, P_W and to the temperature T. By substituting the parameters of λ_S and λ₂, Eq. (13) is simplified as

The experimental conditions are T = 23 °C, P = 1013 mb and RH = 65%. Under these conditions D_S = 1013 mb/296 K and δD_S /D_S = 3 × 10⁻³, the uncertainty of D_S is δD_S = 0.01 mb/K. The variation of air humidity is controlled within ± 3%, and the saturation pressure of water vapor at T = 296 K is P_sat = 28 mb. So the uncertainty of the water vapor density is δD_W < 3 × 10⁻³ mb/K. Therefore, the total uncertainty is | δr/r | < 1.5 × 10⁻⁸.

Suppose the measured displacement Δl is less than 15 µm, then after substituting these uncertainties into Eq. (7), we can get δΔl < 6.67 pm. Suppose Δl is less than 150 mm, then δΔl is about 2.25 nm. These results of δΔl indicate that the LSWI can achieve picometer resolution in the micron range and nanometer resolution in the millimeter range.

5.2 Nonlinearity analysis

The periodic nonlinearity caused by the elliptical polarization, non-orthogonality of polarizing beams and imperfect alignment of polarizing beamsplitters has been an important factor that limits the measurement accuracy at nanometer level in optical interferometers [8–10]. Suppose two beams impinging upon the BS₁ in the LSWI are shown in Fig. 8 . The two electric fields of laser frequencies f₁ and f₂ can be expressed by

 

Fig. 8 Schematic of nonlinearity analysis.

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Here E is the magnitude of electric fields. And ${\phi }_{01x}$, ${\phi }_{01y}$, ${\phi }_{02x}$, ${\phi }_{02y}$ are the original phases of each electric field in x and y directions, ${\phi }_{01x}=4\pi L/{\lambda }_1$, ${\phi }_{01y}=4\pi \left(L+L_2\right)/{\lambda }_1$, ${\phi }_{02x}=4\pi L/{\lambda }_2$, ${\phi }_{02y}=4\pi \left(L+L_2\right)/{\lambda }_2$. L is the optical path difference between M₁-BS₁ and PBS₁-BS₁, L₂ is the optical path difference between PBS₁ and M₂.

For simplicity, we suppose that the optical components in Fig. 1 are ideal and the energy loss during the transmission is negligible. A band-pass filter is used to decouple the DC and high frequency components. Then the filtered signals I_x, I_y received by PD₁ and PD₂ are expressed by

The first terms of I_D1 and I_D2 are the ideal measurement signals in the absence of nonlinearity. The second term indicates that the intensity is wakened in the presence of nonlinearity. And the third term is the factor that distorts the measurement of phase difference between λ₁ and λ₂. Equations (16) and (17) indicate that the nonlinear error in the LSWI is a second-order phase error.

From Eqs. (16) and (17) the following evaluation was made. The phase difference between I_D1 and I_D2 is distorted while α and β are not equal to 0°. Let L = 0, λ₁ = 0.6329906 µm and λ₂ = 0.632992 µm. A simulation to determine the magnitude of the nonlinear error in measuring the displacement was performed with various angular configurations of the BS₁. As shown in Fig. 9 , it shows that the nonlinear error behaves a phase error of one cycle per fringe and the maximum error is about 6 nm as α = 3° and β = 5°, the maximum error is about 1.5 nm as α = 1° and β = 1.5°, and the maximum error is about 0.2 nm as α = 0.5° and β = 0.8°, thus the nonlinear error is sensitive to the angles of α and β.

 

Fig. 9 Simulation of the nonlinearity analysis.

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6. Conclusion

We have developed and experimentally demonstrated a laser synthetic wavelength interferometer compared with an Agilent 5529A interferometer. The measurement principle and the signal processing implement of the LSWI were described in detail. Three displacement experiments were carried out to verify its performance in millimeter, micrometer and nanometer ranges. The results showed that the LSWI is able to measure large displacement with nanometer accuracy. Further, the sources of error influenced on the measurement accuracy and resolution were discussed and evaluated. This showed that the nonlinear error in the LSWI is a second-order phase error per fringe, and the nonlinearity of polarizing beams and the imperfect alignment of polarizing beamsplitter will induce nonlinear error ranging from several nanometers to sub-nanometer that thus must be compensated carefully in practical development.