## Abstract

A laser synthetic wavelength interferometer that is capable of achieving large displacement measurement with nanometer accuracy is developed. The principle and the signal processing method of the interferometer are introduced. The displacement measurement experiments and the comparisons with a commercial interferometer both in small and large ranges are performed in order to verify the performance of the interferometer. Experimental results show that the average errors and standard deviations of the interferometer are in accordance with those obtained from the commercial interferometer. The resolution and the nonlinearity of the interferometer are also discussed in detail. These results show that the development of the interferometer is reasonable and feasible.

©2010 Optical Society of America

## 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 λ_{1} and λ_{2} 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 λ_{1} is split into two beams with almost equal intensity by a beamsplitter (BS_{1}), one directed to the reference corner cube (M_{1}) and the other to a polarizing beamsplitter (PBS_{1}). The reflected beams are recombined in BS_{1} and reflected by another polarizing beamsplitter (PBS_{2}), and then the interference signal is detected by a photodetector (PD_{1}). Similarly, the beam λ_{2} is split into two beams with almost equal intensity by BS_{1}, one directed to M_{1} and the other transmitted through PBS_{1} to a moving corner cube (M_{2}). They are reflected back, get together in BS_{1} and then pass through PBS_{2}. This interference signal is split by another beamspliter (BS_{2}) into two signals detected by PD_{2} and PD_{3}, respectively. PD_{2} and PD_{3} 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 PD_{1} and PD_{2} and the bidirectional fringe counting with PD_{2} and PD_{3} are completed in the signal processing unit. When M_{2} is moved a measured nanometer displacement Δ*l* less than half a wavelength, then the signal processing unit sends a signal to the stage controller, so M_{1} is moved a compensative displacement Δ*L* in order to keep the phase difference constant. Through theoretical derivation [11,12], the measured displacement Δ*l* expressed by

This shows that a smaller displacement Δ*l* to be measured can be realized by determining a larger displacement Δ*L* because the synthetic wavelength λ_{S} is much greater than the single wavelength λ_{2}. From Eq. (1), suppose the wavelength λ_{2} = 632.8 nm and the synthetic wavelength λ_{S} = 280 mm, it is obvious that 1 nm measured displacement of M_{2} corresponds to 0.44 mm displacement of M_{1}, and 0.01 nm measured displacement of M_{2} to 4.4 µm displacement of M_{1}. That is, if we measure 0.44 mm or 4.4 µm displacement of M_{1}, we can achieve 1 nm or 0.01 nm measured displacement of M_{2}. 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 M_{2} is given by

*N*is the integer fringe number of beam λ

_{2}.

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 M_{1}.

#### 3.1 Interference signals processing

As shown in Fig. 2
, the curves *V(λ _{1})*,

*V(λ*)

_{2}*cosine*and

*V(λ*represent the interference signals received by PD

_{2})sine_{1}, PD

_{2}and PD

_{3}, respectively. The curves

*V(λ*and

_{1})*V(λ*)

_{2}*cosine*are used to determine the phase difference between the interference signals λ

_{1}and λ

_{2}, and the

*V(λ*)

_{2}*cosine*and

*V(λ*are used to count the integer fringe number of beam λ

_{2})sine_{2}and to judge the moving direction of M

_{2}. The filling pulse number n

_{0}represents the original phase difference before M

_{2}starts moving. The dashed curve

*V'(λ*signifies the signal when M

_{2})cosine_{2}is moved a certain displacement corresponding to a fractional fringe, the filling pulse number n

_{1}signifies current phase difference. To keep the phase difference constant, M

_{1}is moved forward or backward according to the moving direction of M

_{2}. Thus, n

_{1}approaches to n

_{0}. In addition, when M

_{2}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(λ*is invariable during the moving of M

_{1})_{2}, thus the signal

*V(λ*can be regarded as a reference signal in order to realize the combination of fraction and integer fringes.

_{1})#### 3.2 Compensation of air refractive index

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

_{20}and λ

_{S0}are the laser wavelengths in the vacuum,

*n(λ*is the air refractivity λ

_{2})_{2}during the moving of M

_{2}.

*n'(λ*,

_{2})*n'(λ*are the air refractive indices when the M

_{S})_{1}is moving.

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 CO_{2} content is negligible, the revised dispersion formula for standard condition air at 1 Pa and 15 °C is

*T*and pressure

*P*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 PD_{1} and PD_{2} are filtered, amplified and converted to square waves to realize the determination of the phase difference. Interference signals of λ_{2} detected by PD_{2} and PD_{3} 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 M_{1} according to the phase difference variation.

## 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 M_{1} displacement Δ*L* 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 M_{2} 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.

As the experiment began, firstly, the M_{1} was moved to find a position where the phase difference between the two interference signals λ_{1} and λ_{2} was set an original value of zero. Secondly, the M_{2} was moved a measured displacement Δ*l*, the integer fringe number *N* was calculated by the DSP. Thirdly, the M_{1} was moved forward or backward according to the moving direction of M_{2} to make the phase difference reach to the original value of zero again, the M_{1} displacement Δ*L* was recorded. Substituting *N* and Δ*L* into the Eq. (2), the measured displacement Δ*l* 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 M_{1} 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.

#### 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.

#### 4.3 Displacement measurement experiment in nanometer range

In this experiment, the M_{2} 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.

## 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 *δ*λ_{2} represents the uncertainty of single wavelength, *δ*λ_{S} represents the uncertainty of synthetic wavelength, *δ*Δ*L* is the positioning accuracy of the stage and the *δr/r* takes into account the errors of Δ*l* induced by the uncertainty of ratio *n(λ _{S})*/

*n(λ*of the air refractive induces at the two wavelengths λ

_{2})_{S}and λ

_{2}. The maximum displacement of Δ

*L*is λ

_{S}/2.

According to the parameters provided by the ML10 laser manual, the frequency *f _{1}* = 473612829 MHz and

*f*= 473611744 MHz. The frequency stability is 0.05 ppm. So the variations of

_{2}*δ*λ

_{2},

*δ*λ

_{S}can be calculated as

The positioning accuracy of the stage *δ*Δ*L* is determined mainly by the moving velocity of the stage and the frequency of fast clock shown in Fig. 2. The *δ*Δ*L* 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

*δ*Δ

*L*= 2.91µm

Basing on the researches of Owens and Castell [14,15], the uncertainty of ratio *n(λ _{S})*/

*n(λ*can be expressed by

_{2})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% CO

_{2}) and

*D*of water vapor are related to the partial pressures

_{W}*P*,

_{S}*P*and to the temperature

_{W}*T*. By substituting the parameters of λ

_{S}and λ

_{2}, 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*= 3 × 10

_{S}/D_{S}^{−3}, the uncertainty of D

_{S}is

*δD*= 0.01 mb/K. The variation of air humidity is controlled within ± 3%, and the saturation pressure of water vapor at

_{S}*T*= 296 K is

*P*= 28 mb. So the uncertainty of the water vapor density is

_{sat}*δD*< 3 × 10

_{W}^{−3}mb/K. Therefore, the total uncertainty is |

*δr/r*| < 1.5 × 10

^{−8}.

Suppose the measured displacement Δ*l* is less than 15 µm, then after substituting these uncertainties into Eq. (7), we can get δΔ*l* < 6.67 pm. Suppose Δ*l* is less than 150 mm, then δΔ*l* is about 2.25 nm. These results of δΔ*l* 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 BS_{1} in the LSWI are shown in Fig. 8
. The two electric fields of laser frequencies *f _{1}* and

*f*can be expressed by

_{2}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 M_{1}-BS_{1} and PBS_{1}-BS_{1}, *L _{2}* is the optical path difference between PBS

_{1}and M

_{2}.

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*received by PD

_{y}_{1}and PD

_{2}are expressed by

The first terms of *I _{D1}* and

*I*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 λ

_{D2}_{1}and λ

_{2}. 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*is distorted while α and β are not equal to 0°. Let

_{D2}*L*= 0, λ

_{1}= 0.6329906 µm and λ

_{2}= 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 BS

_{1}. 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.

## 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.

## Acknowlegement

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 50827501 and No. 90923026, the Zhejiang Provincial Natural Science Foundation of China (ZJNSF) under Grant No. Y107556 and the Research Foundation of the Department of Education of Zhejiang Province, China under Grant No. ZD2007004 and YK2008052.

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