## Abstract

A method based on a specific quasi-common-optical-path (QCOP) configuration for two-dimensional displacement measurement is presented. The measurement system consists of a heterodyne light source, two-dimensional holographic grating, specially designed set of half wave plates, and lock-in amplifiers. Two measurement configurations, for single and differential detection, are designed. The sensitivity, resolution and nonlinear phase error of the differential detection type are better than those of the single detection type. The experimental results demonstrate that the QCOP interferometer has the ability to measure two-dimensional displacement while maintaining high system stability.

©2011 Optical Society of America

## 1. Introduction

Displacement related measuring techniques play an important role in modem technology being widely applied in many fields, such as in the semiconductor industry, precision manufacturing, photolithography, metrology instrument, high density mass storage systems, to only named a few [1–5]. These types of measurement systems have been developed to such a stage that they are less and less restricted to just the micro-scale resolution but can now achieve nano-scale resolution, as well as meso-scale measurements. Therefore, much attention is being paid to develop more precise measuring devices or methods with the ability to provide exact displacement information with high precision. Various tools to achieve precise measurement have been developed, such as the capacitance sensor, linear encoder, strain gauge or interferometer [6–11]. The resolution of such measuring sensors can reach the nanometer, even the sub-nanometer level. Among them, one of the most common instruments is the capacitance sensor, which is often used because of its high measuring resolution, compact size and low cost. In general, the resolution of the capacitive sensor can achieve sub-nanometer precision [7]. Unfortunately, the measuring range of this sensor type is limited to 100 micrometers, which is not enough for meso-scale displacement measurement.

In contrast, linear encoders are inexpensive displacement measuring sensors that have been adapted for displacement measuring with high resolution. Among the various types of linear encoders [8,9], moire encoders are most commonly used because they are relatively easy to manufacture and offer a high resolution-per-cost ratio. However, a measurement error can be caused by an unwanted assembly tilt angle between the two scales. It is difficult to eliminate this unwanted tilt [9]. Since the linear encoder can only be used to obtain position information on the assembled axis, it is much more suitable for one-dimensional displacement measurement. For example, for two-dimensional (2D) applications, when two linear encoders are implanted into an XY motion system, any fabrication error in the motion system will result in a motion error which appears to be orthogonal to the proposed moving path. This is the shortcoming of the linear encoder for 2D applications.

Laser interferometers are very useful for displacement measurement. They not only have high resolution and wide dynamic measurement range, but allow for a flexible arrangement of optical paths and direct linkage to the length definition. There are many kinds of interferometer designs which have demonstrated nanometer, even sub-nanometer precision for long measurement ranges [11,12]. Normally, a pair of laser interferometers mounted perpendicular to each other can be used for 2D displacement measurement. However, it is essential for the two interferometers to be arranged orthogonal to each other to ensure measurement accuracy. The additional expense of another interferometer is another disadvantage for t2D measurement. Moreover, due to the fact that most interferometers are based on a non-common optical path (NCOP) design [10,13,14], it is difficult to avoid the problem of environmental disturbance arising from temperature fluctuations, air drift, etc. Thus if one considers that most high precision applications must be carried out in an ambient atmosphere (air), one of the most difficult design tasks is to make a system that is insensitive to disturbing variations. Much attention is being paid to finding methods for 2D displacement measurement that not only ensures high accuracy but also great stability.

In a previous study, we proposed a heterodyne grating interferometer based on a quasi-common-optical-path (QCOP) configuration for displacement measurement [11]. The QCOP configuration means that the measurement and reference beams of the interferometer have almost equivalent optical paths. Surrounding disturbances can be compensated for by this configuration, making the system less sensitive to environmental disturbances. However, the previous QCOP configuration was only for one-dimensional displacement measurement. In this study, we propose an innovative 2D QCOP heterodyne grating interferometer for 2D displacement measurement. By inserting two specific semicircular HWPs into a heterodyne grating interferometer, the movement of a XY motion stage can be easily and precisely measured. The QCOP configuration of the interferometer leads to an effective improvement in the system stability. The device still has the ability to measure 2D displacements simultaneously. The theoretical basis for the 2D QCOP method is described in detail below. Three major tests are performed to demonstrate the ability of the 2D QCOP heterodyne interferometer.

## 2. Theoretical basis

#### 2.1 Heterodyne light source

As shown in Fig. 1
, the light beam coming from a laser source is linearly polarized at 45° with respect to the *x*-axis. The fast axis of electro-optic modulator (EOM) is along the *x*-axis. According to Su’s principle [10], when a linearly polarized beam passes through the EOM, which is driven by an external sawtooth half-wave-voltage with frequency *ω*, the complex amplitude of the light beam can be described as follows [11]:

The light beam is then enlarged using a beam expander. Two semicircular half-wave plates (HWPs) for which the fast axes are at 45° are inserted. These two semicircular HWPs (HWP1, HWP2) are arranged as in Fig. 1. After passing through the two HWPs, the expanded heterodyne beam is divided into 4 parts *A*, *B*, *C* and *D*. According to the Jones calculation, the amplitudes of these 4 parts are given by

*J*(180°) and

*J*(0°) are the Jones matrices of different phase retardations in the different quarters of HWPs.

#### 2.2 Phase difference measurement with QCOP heterodyne grating interferometry

As shown in Fig. 2
, after being reflected from the mirror, the expanded heterodyne light beam is focused on the 2D holographic grating by a lens and then diffracted. By means of choosing a suitable focusing lens and grating pitch, the zero order (*m* = 0) beam will overlap with the plus and minus first order (*m* = 1) diffracted beams. When the grating moves with a displacement (*l _{q}*) along the

*q (q = x or y)*direction, the relationship between the displacement (

*l*) and the optical phase

_{q}*ϕ*of the

_{qm}*m*

^{st}order diffracted beam can be written as

*p*is the grating pitch and

*m*is the diffraction order. The beam distribution is shown in detail in the inset. For example,

*A*

_{x1}and

*A*

_{y-1}represent the parts

*A*of the plus and minus first order diffracted beam on the

*x*and

*y*axis, respectively, and so on. As shown in the inset, there are eight overlapping areas numbered from O

_{1}to O

_{8}. Thus O

_{5}represents the overlap of the part

*C*of the 1st order (

*C*

_{y}_{1}) with the part

*A*of the 0 order (

*A*

_{0}). Correspondingly, O

_{1}represents the overlap of the part

*B*of the 1st order (

*B*

_{x}_{1}) with the part

*A*of the 0th order (

*A*

_{0}). It is worth noting here that when the grating moves along the

*x*direction, the interference phase changes can be observed from the overlapping area O

_{1}to O

_{4}. In the same way, when the grating moves along the

*y*direction, the interference phase changes can be observed from the overlapping area O

_{5}to O

_{8}. This means that this configuration provides four overlapping areas in each direction for measuring the changes in the interference phase.

The overlapping areas of O_{1}~O_{8} depend on the solid angle of the focusing beam and the grating pitch, and can be calculated by grating equation as *p*(sin*θ*
_{in} + sin*θ*
_{diff}) = *m*λ. One can use an iris before the focus lens to turn the solid angle. In our case, the overlapping areas of O_{1}~O_{8} are about 0.015(radian). Moreover, two of the overlapping areas (O_{1} and O_{5}) are chosen to pass through two polarizers P_{1} and P_{2} with transmittance axes at 0°. The interference of the light is detected using two detectors D_{1} and D_{2}. The interference signal *I*
_{1} measured by the detector D_{1} can be written as

*d*stands for the optical path difference between light that goes through the HWP, or not. We ensure that

*ϕ*

_{x}_{0}= 0 and that the optical path difference (Δ

*d*) can be assumed to be a constant and thus can be ignored. Furthermore, no matter what the distance between grating and detector is, the interference signals of the overlapping areas won’t be affected by other diffraction beams.

Similarly, the interference signal *I*
_{2} measured by the detector D_{2} can be written as

_{3}for which the transmittance axis is at 45° and a detector D

_{3}are used to measure the intensity of the non-overlapping areas of

*A*

_{0},

*B*

_{0},

*C*

_{0}, or

*D*

_{0}. The heterodyne signal

*I*

_{3}measured by D

_{3}can be written asThen, all three signals

*I*

_{1},

*I*

_{2}(measurement signals) and

*I*

_{3}(reference signal) are sent into two lock-in amplifiers. The phase differences

*Φ*=

_{x}*ϕ*

_{x}_{1}(between

*I*

_{1}and

*I*

_{3}) and

*Φ*=

_{y}*ϕ*

_{y}_{1}(between

*I*

_{2}and

*I*

_{3}) are given bywhere

*q*stands for

*x*or

*y*. From Eq. (7), it is obvious that the grating displacements (

*l*) in the

*x*or

*y*direction can be calculated based on measurement of the phase difference variations (

*Φ*and

_{x}*Φ*) and the grating pitch (

_{y}*p*)

In addition, due to fact that the optical paths of the orthogonally polarized beams in the overlapping areas (O_{1}) and (O_{5}) are almost equivalent, they will suffer from similar surrounding disturbances. These environmental disturbances will be compensated for in the interference signal. We regard this design as a 2D single type QCOP configuration.

#### 2.3 Single and differential type QCOP heterodyne grating interferometry

It is worth mentioning here that the configuration of the QCOP method can be further classified into two types. The one described above is a single type QCOP configuration while the other is a differential one. Figure 3
shows the schemes of the single and differential type QCOP methods. Two polarization beam splitters (PBSs) are used to separate the two overlapping beams into four parts. According to the optical arrangement and Jones calculation, the interference signals *I*
_{4} and *I*
_{5} measured by the detectors D_{4} and D_{5} can be written as

*I*

_{6}and

*I*

_{7}measured by detectors D

_{6}and D

_{7}can be written as

*I*

_{4}and

*I*

_{5}for the

*x*direction) and (

*I*

_{6}and

*I*

_{7}for the

*y*direction) are sent into two phase meters. The phase differences

*Φ*=

_{x}*ϕ*

_{x}_{1}– (–

*ϕ*

_{x}_{1}) (between

*I*

_{4}and

*I*

_{5}) and

*Φ*=

_{y}*ϕ*

_{y}_{1}– (–

*ϕ*

_{y}_{1}) (between

*I*

_{6}and

*I*

_{7}) are now given as

The differences between the single and differential type QCOP heterodyne grating interferometers will be further discussed later.

## 3. Experimental setup and system performance

#### 3.1 Experimental setup

To demonstrate the feasibility of our 2D QCOP heterodyne grating interferometer, the displacement provided by a motion stage was measured. The experimental configuration is depicted in Fig. 4
. The red line in Fig. 4 represents the optical path for the single type 2D QCOP method. This method relies on a heterodyne light source composed of a linearly polarized He-Ne laser light modulated by an electro-optic modulator EOM (model: 4001, New Focus, Inc.). The frequency difference between the *p*- and *s*- polarizations of the heterodyne light source is 20 kHz. A 2D diffraction grating with a pitch of 3.2 μm is mounted on a precision double layer XY stepper (model: XYS-50; Measure control, Inc.). A lens with a focal length of *f* = 25.4 mm, was chosen to focus the beam on the grating. The overlapping areas of the diffracted beams are reflected by a mirror. The interference signals are measured after passing through three polarizers and three detectors (model: PDA-36 EC, Thorlabs, Inc.). The contrast of the detection signals, the ratio of DC to AC, in our experiment is about 25%. However, because of the heterodyne detection, the signals with this low contrast are enough for phase measurement. Then, two lock-in amplifiers (model: SR850, Stanford Research Systems, Inc.), with an angular resolution of 0.001° are used to measure the phase difference of the two interference signals in each direction.

#### 3.2 System performance

In order to demonstrate that our QCOP method has the ability to measure millimetre-scale range displacement with a nanometre resolution, an XY stepper with two integrated linear encoders (LEs) was used in a closed-loop configuration for positioning. First, we asked the XY stepper to move forward and backward (1D W motion) with a displacement of 1 mm along the Y direction. A commercial linear interferometer (model: HP 5529A) was used to simultaneously measure the stepper movement. Measurement results obtained by QCOP method and the commercial interferometer are shown in Fig. 5 . As shown in Fig. 5(a), the two curves show almost the same displacement and behavior, it is worth noting that the measurement curves obtained using our method (blue, solid line) are as linear as those obtained using the commercial linear interferometer (red, solid line). This demonstrates that the QCOP method has the ability to measure 1D displacement as a commercial linear interferometer. Besides, Fig. 5(b) shows the measurement deviation between our method and the HP interferometer for the Path 1 measurement. One can found that the deviation is about 1.6 μm. We believe that the discrepancy may result from the fact that the test beam of HP interferometer might not exactly coincide with the moving direction of grating or Abbe error In addition, the grating of our QCOP method and the retroreflector of the HP interferometer were not installed at the same position on the XY stepper, and this situation also would cause slightly measuring difference.

In order to test the potential of the QCOP method for 2D displacement measurement, quadrangle motion was performed for a 1 mm displacement in the X and Y directions. Since this commercial linear interferometer only can be used to measure 1D displacement in this study, we adopt the LEs to sense the displacement in their working axis to conform to the measured displacement obtained by our 2D QCOP method. The LE of the X axis of the stepper was independent of the Y axis. The moving path is classified into four parts (Path 1~4). Experimental data are shown in Figs. 6 and 7 . Figure 6 shows the results in three dimensions. It can be seen that the trend and behavior of the displacement measured by the 2D QCOP method (QCOP, red, dashed line) is conformed to the displacement measured by the LE (LE, green, short dots). However, the experimental data show that a slight angle exists between the moving direction and the grating, which will cause measurement deviation between the QCOP and LE. Figure 7 shows a top view of the experimental results in the XY section. It can be seen from the local enlargement area, that when the XY stepper moves with a displacement of 1mm in the X direction, the 2D QCOP method measured 12 μm in the Y direction. The angle of misalignment was about 0.688°. There are two ways to eliminate this misalignment angle. One is to use a precisely rotating stage to adjust the angle between the grating and the XY stepper. The other is to use coordinate transformation to remove the effects caused by this misaligned angle. The curve of the QCOP (adjusted) (blue, solid line) shows the experimental results without misalignment angle effects. The difference in displacement in the X direction of the path 1 measured by our QCOP method and LE is 28.7 nm for a displacement of 1 mm. This confirms that the QCOP method is able to measure 2D (1 mm × 1 mm) displacement as a commercial linear encoder.

Moreover, it is worth mentioning here, by subtracting our measuring data from an ideal path, as one can find from Fig. 8(a) , that the moving trajectory of the path 1 measured by our 2D QCOP method is not a straight line. Clearly, there exists a lateral deviation of 0.45 μm in the Y direction when the XY stepper moves along the X direction for the path 1. This deviation is so-called straightness error, and can only be measured by our 2D QCOP method. This is because the two LEs cannot provide straightness information in their working directions but our 2D QCOP can. As shown in Fig. 8(b), we also found that there is a lateral deviation of 2.2 μm in the X direction when the XY stepper moves along the Y direction for the path 2.

Furthermore, the XY stepper was asked to move along an octagonal path (Path 1~8). The experimental results are shown in Fig. 9(a) . Clearly, the shapes of the two measurement curves, adjusted QCOP and LE are the same, which is in agreement with the fact that our method is able to measure 2D displacement over millimeter range. Besides, by subtracting our measuring data from an ideal path, the results are shown in Fig. 9(b). Figure 9(b) shows there is a lateral deviation of 0.55 μm in the Y direction when the XY stepper moves along the X direction (Path 1). Obviously, the real moving trajectory of the Path 1 can only be obtained by our 2D QCOP method. These above results demonstrate that our 2D QCOP method has the ability to measure 2D displacement and straightness simultaneously without reorganizing the optical setup. In addition, the maximum measurement range depends on the valid area of the 2D holographic grating. The one used in the experiments was about 10 mm × 10 mm. A large-area grating can be used to increase the measurement range.

## 4. Discussion

There are two types of system configurations for the QCOP heterodyne grating interferometer. One determines which architecture is suitable for their own applications by analyzing their status with the following conditions.

#### 4.1 Measurement resolution and sensitivity

From Eq. (7) and Eq. (11) it can be clearly seen that the resolution of differential type QCOP heterodyne grating interferometer is twice that of the single type. This also means that the sensitivity of the differential type QCOP interferometer is greater than the single type. If the phase resolution (0.001°) of the lock-in amplifier is considered, the corresponding displacement resolutions of the differential and single type interferometers are estimated to be 9 pm and 4.5 pm for a grating pitch of 3.2 μm, respectively. However, from our experimental results, we observe that if only high frequency noise is considered, the measurement resolution of the differential and single type QCOP interferometers can be estimated to be 1.41 nm and 2.52 nm. We believe that the resolution is influenced by the problem of random noise, which mainly comes from the laser source, photodetectors and the electronics of the lock-in amplifiers. There are two possible ways to further improve the resolution. For example, increasing the time constant of the lock-in amplifier would attenuate the high frequency noise level. Therefore, one solution would be to increase the time constant of the RC filter in the lock-in circuit. Another solution would be to decrease the grating pitch [13]. However, the overlapping area also would be decreased when decreasing the grating pitch. Therefore it is suggested that the differential type of solution should be adopted when one needs much higher measurement resolution.

#### 4.2 Non-linear error

It is worth discussing the non-linear periodic error of heterodyne detection in the QCOP method. Actually, the error sources mainly come from misalignment of the HWP and polarizer (frequency mixing), as well as leakage from an imperfect polarizer (polarization mixing) and HWP (polarization-frequency mixing). To estimate the error resulting from the above mentioned effects, the expression of the interference amplitude can be written as

*θ*

_{p}is the azimuth angle of the polarizer,

*ε*is the alignment error of azimuth angle of the HWP,

*α*is the extinction ratio of the imperfect polarizer, and

*δ*is the phase retardation error introduced by an imperfect HWP. The interference signal can be written as follows:where

*Φ*' is the deformed phase. Thus, the phase error of the single type QCOP configuration can be obtained by

In our current experiment setup (case 1), the minimum adjusting angles of polarizers and HWPs could be controlled to within 5′ and 25″ by precision rotation mounts. The extinction ratio of the polarizers and phase retardation error of HWPs are 0.0001 and 1.2°. By substituting the above parameters (*α* = 0.0001, *θ*
_{0} = 0°5′, *θ _{9}*

_{0}= 90°5′,

*ε*= 25″ and

*δ*= 1.2°) into Eq. (14) and Eq. (15), the non-linear periodic error of the differential and single types are estimated to be about 2.4° and 1.2°. The corresponding displacements of 10.7 nm can be further obtained by Eq. (7) and Eq. (11). The simulation results of the non-linear periodic error of the differential and single types are shown in Fig. 10 . However, when the azimuth angles of the polarizer or the HWP increases, such as

*θ*= 47° (in case 2), the corresponding non-linear periodic errors for the differential and single type interferometers are 2.6° and 8.1°, and the corresponding displacements are 11.6 nm and 72 nm, respectively. Thus it can be seen that when the azimuth angle errors of the HWP or the polarizer increase, the influence of non-linear error on differential type is much less than that on single type. Clearly, the differential type QCOP interferometer is suggested to avoid the influence coming from the non-linear periodic error. In our experiment, we found real non-linear errors of about 10 nm for every grating pitch in displacement. Actually, there are several ways to eliminate the non-linear periodic error [12]. One of the methods is to reduce the period of the grating pitch. For example, by replacing the period of the grating pitch of 3.2 μm with 1.67 μm, the corresponding non-linear periodic error of the differential type can be down to 6 nm, while that of the single type can be decreased to 37.6 nm. Another way of eliminating non-linear periodic error is to use a symmetric design [12].

_{h}## 5. Conclusion

We propose an innovative 2D QCOP heterodyne grating interferometer for 2D displacement measurement. The measurement system includes a heterodyne light source, a 2D holographic grating, half wave plates, and lock-in circuits. By inserting two specific semicircular HWPs into a heterodyne grating interferometer, 2D displacement of a motion stage can be easily and precisely obtained. Based on the design of the QCOP method, the system can be classified as either the single or differential type. The difference between the single and differential QCOP configurations have been discussed and analyzed. The sensitivity, resolution and nonlinear phase error of the differential type QCOP interferometer are better than the single type. However, the single type QCOP interferometer is much easier to set up and perform. Using the QCOP configuration, the interferometer can effectively improve the system stability while still having the ability to measure meso-scale displacements simultaneously. The experimental results demonstrate that our method has the ability to measure 2D displacement wile still maintaining high system stability. Further developments lie in 2D displacement feedback implementation.

## Acknowledgment

The authors cordially thank Mr. Mini-Pei Lu from National Central University, Taiwan, Mr. Regis Deturche from the University of Technology of Troyes, France, and assistant Prof. Chyan-Chyi Wu from Tamkang University, Taiwan for their assistance. This study was supported by the National Science Council, Taiwan, under contract NSC 97-2221-E-008-022-MY3 and the Orchid Program (18256SC). One of the authors, H.-L. Hsieh, would like to thank the IFT (Institut Français de Taipei) for PhD scholarship support.

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