Abstract

A grating-based interferometer for 6-DOF displacement and angle measurement is proposed in this study. The proposed interferometer is composed of three identical detection parts sharing the same light source. Each detection part utilizes three techniques: heterodyne, grating shearing, and Michelson interferometries. Displacement information in the three perpendicular directions (X, Y, Z) can be sensed simultaneously by each detection part. Furthermore, angle information (θX, θY, θZ) can be obtained by comparing the displacement measurement results between two corresponding detection parts. The feasibility and performance of the proposed grating-based interferometer are evaluated in displacement and angle measurement experiments. In comparison with the internal capacitance sensor built into the commercial piezo-stage, the measurement resolutions of the displacement and angle of our proposed interferometer are about 2 nm and 0.05 μrad.

© 2015 Optical Society of America

1. Introduction

Over the past few years, displacement measurement techniques have come to be broadly utilized in the semiconductor industry, precision machinery industry, microscopic techniques, optical alignment systems, and medical research. For further development of these fields, there is a strong need for high precision measurement techniques providing high resolution, high stability, and large measurement range. For example, large-area surface topography can be investigated using a combination of atomic force microscopy with a precision displacement measurement device and motion stage. However, the accuracy of the measurement device will affect the measurement result obtained by the atomic force microscope. This shows the importance of the precision displacement measurement technique. Many researchers have attempted to develop precision displacement measurement techniques with the ability to provide exact displacement information in order to extend applications in the fields mentioned above.

Laser interferometry is one of the most useful displacement measurement techniques due to its advantages of long measurement range, high measurement resolution, and flexible arrangement [1–5]. The first commercial laser interferometer was developed by the Hewlett-Packard company in 1974 which combined a dual frequency light source with a Wollaston prism, and could be used to measure one-degree-of-freedom (1-DOF) displacement. From that date, various interferometers have been designed to achieve nanometric level resolution [6–11]. In addition to 1-DOF displacement measurement, techniques for corresponding DOF measurement are essential for 2-DOF or multi-DOF applications [12–15].

Many studies have been carried out to find ways to meet the requirements for measuring 2-DOF or multi-DOF displacement precisely [16–19]. In general, more than one-DOF displacement measurement can be achieved by adding another interferometer unit, using beam dividing techniques, or replacing the reflective mirrors of the interferometer with a diffraction grating. Chen et al. [16] proposed a heterodyne interferometer design using a Zeeman laser for 2-DOF displacement measurement. However, 2-DOF displacement information could not be obtained simultaneously without changing the optical configuration. Gao et al. developed a heterodyne grating-based interferometer that used a planar diffraction grating for 2-DOF displacement measurement [17]. 2-DOF in-plane displacement could be measured simultaneously using this optical configuration. Gao’s team further modified their grating-based interferometer to obtain the 3-DOF displacement behaviors of a motion stage [18]. To achieve multi-DOF displacement measurement, the optical configuration of the interferometer should be composed of two or more detection sections. For example, Li et al. [19] developed a 6-DOF grating-based interferometer sharing the same light source. 6-DOF measurement has been realized by combining a three-axis displacement sensor and a three-axis autocollimator in a simple manner.

In a previous study, we proposed a heterodyne grating-based interferometer based on the quasi-common optical path (QCOP) design for 2-DOF in-plane displacement measurement [20]. The optical paths between the reference and measurement beams are almost equivalent. Surrounding disturbances can be compensated for by this technique, making the laser interferometer less sensitive to environmental disturbances. However, the technique was only useful for 2-DOF displacement measurement. In this paper, an innovative heterodyne laser interferometer for 3-DOF and 6-DOF measurements is described. The proposed technique combines the merits of the heterodyne, Michelson, and grating-shearing interferometries. According to the measurement principle, 2-DOF in-plane and 1-DOF out-of-plane displacement information (X, Y, Z) and 3-DOF rotation angles (θX, θY, θZ) can be measured simultaneously without changing the optical configuration. The measurement principle of our proposed grating-based interferometer is described in detail below. After introducing the principles on which this technique is based, its feasibility is demonstrated.

2. Measurement principles

The optical configuration of our proposed grating-based interferometer is illustrated in Fig. 1. For convenience, the Z-axis is selected to be along the direction of light propagation, and the Y-axis is along the horizontal plane. The light beam coming from a laser source is linearly polarized at 45° with respect to the X-axis. The heterodyne light source is obtained by using an electro-optic modulator (EOM) for amplitude modulation. According to Su’s modulation principle [21], the complex amplitude of the heterodyne light beam can be written as follows:

EEOM=ei(kl+ϕ)(eiωt/2eiωt/2),
where ω is the modulation frequency of the light source, k indicates the wave number, ϕ stands for the initial phase of the light beam, and l means the optical path.

 

Fig. 1 System configuration for 3-DOF displacement measurement.

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After obtaining the heterodyne light source, a semi-transmission type grating with a two-dimensional structure is used to construct two detection configurations for measuring the in-plane and out-of-plane displacement information. The out-of-plane (Z-axis) and in-plane (X- and Y-axis) displacements are measured by using Michelson detection and the grating-shearing detection configurations, respectively. The principles of the proposed interferometer for displacement measurement will be described below.

2.1 2-DOF In-plane displacement measurement (X- and Y-axis)

The in-plane displacement detection configuration is framed by a blue dashed line in Fig. 1. The working principle is based on the QCOP configuration [20]. The heterodyne light beam is expanded by a beam expander. Two semicircular half-wave-plates (HWP1, HWP2) are placed in front of the semi-transmission type grating with the fast axis at 45°. After passing through the two half-wave-plates, the expanded heterodyne beam is divided into 4 parts (Parts A, B, C, and D). According to the Jones calculation, the electric-fields of 4 parts are given by

A=D=J(0)EEOM(eiωt/2eiωt/2),B=C=J(180)EEOM(eiωt/2eiωt/2),
where J(0°) and J(180°) stand for the Jone’s matrixes of the HWP1and HWP2 [20]. The heterodyne light beam with four polarization status is focused on the transmission type grating by using a focusing lens and then diffracted. By choosing a suitable focusing length and grating pitch, the plus and minus first orders and zeroth order of the diffraction wavefronts interfere to generate the heterodyne signals. When the grating moves along the in-plane direction with a displacement (lq, q = X or Y), the optical phase ϕqm of the mst order of the diffracted beam will vary. The relationship between the displacement and the optical phases can be written as
ϕqm=2mπlq/p,
where p and m stand for the grating pitch and the diffraction order, respectively. The beam distribution is shown in detail in the inset to Fig. 1. For example, AX+1 and AY-1 represent the parts A of the plus and minus first order diffracted beams on the X and Y axis, respectively, and so on.

In order to retrieve the three interference beatings carried by each polarization, three polarizers, P1, P2 and P3, with transmittance axes at 45°, 0° and 0° are placed in front of detectors D1, D2 and D3. With this optical arrangement and according to the Jones calculation, the AC term of the two interference signal I2 and I3 detected by D2 and D3 will carry the optical phase difference of the 1st and 0th order diffracted beams in the X- and Y-directions and can be formulated as

I2|P2(0)(BX0+DX+1)|2cos[ωt+(ϕX0ϕX+1)].I3|P3(0)(BY0+AY1)|2cos[ωt+(ϕY0ϕY1)]
Here, we ensure that ϕX0 = ϕY0 = 0. Similarly, the interference signal I1 detected by D1 can be written as
I1cos(ωt).
The reference signal (I1) and measurement signals (I2 and I3) are sent into two lock-in amplifiers. The phase differences (ΔΦq, q = X or Y) between I1 and I3 and between I2 and I3 are given by
ΔΦq=2πΔlq/p,
It is obvious that the in-plane displacements (lq) can be calculated based on the measurement of the phase difference variations and the grating pitch (p)

Δlq=pΔΦq/2π.

2.2 1-DOF Out-of-plane displacement measurement (Z-axis)

The detection configuration for out-of-plane displacement is framed by a yellow dashed line in Fig. 1. The semi-transmission grating functions as a reflective mirror in this case. After passing through a beam-splitter (BS), the heterodyne light beam is divided into two. One beam is reflected by the BS to a stationary mirror and then back to detector D4. The reflected beam is regarded as the reference beam. Meanwhile, the transmitted beam moves toward the semi-transmission grating to be reflected back to detector D4. The transmitted beam is called the measurement beam. The electrical fields of the reference beam (Er) and measurement beam (Em) can be formulated as

Er=ei(klr+ϕr)(eiωt/2eiωt/2),Em=ei(klm+ϕm)(eiωt/2eiωt/2),
where lr and lm stand for the optical paths of the reference and measurement beams; ϕr and ϕm indicate the initial phases of the two beams. As we can see, this is a typical detection configuration for Michelson interferometry. The out-of-plane displacement of the grating can easily be detected using Michelson interferometry. A polarizer with transmittance axes at 45° is placed in front of detector D4 to obtain the interference signal (I4). The optical phases of ϕr and ϕm can be ignored in this situation. The interference signal can be written as
I4=|P4(45)(Er+Em)|2cos[ωt+k(lrlm)].
After sending interference signals I1 and I4 into the lock-in amplifier, the phase difference (ΔΦZ) is given by
ΔΦZ=k(lrlm)=2ΔlZ.
where ΔlZ means the variation of the out-of plane displacement of the grating on the Z-axis. From Eq. (10), the out-of-plane displacement of the grating on the Z-axis can be written as
ΔlZ=ΔΦZ2k=ΔΦZλ4π,
Clearly, the 3-DOF displacement information (X, Y, Z) can be obtained simultaneously without changing the optical configuration.

2.3 3-DOF angle measurement (θX-, θY-, and θZ- axis)

As shown in Fig. 2, the optical configuration for 3-DOF displacement measurement is further modified into three detection parts (I, II, and III) sharing the same light source. Each detection part can be used to detect 3-DOF displacement information. For example, in detection part II (framed by the blue dashed line), using this optical arrangement and the Jones calculation, the interference signals I5, I6 and I7 measured by detectors D5, D6 and D7 are formulated as follows:

 

Fig. 2 Optical configuration for 6-DOF measurement.

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I5cos[ωt+(ϕX0IIϕX+1II)]I6cos[ωt+(ϕY0IIϕY1II)]I7cos[ωt+k(lrIIlmII)]

In detection part II, the in-plane displacement information in the X- and Y-directions can be obtained by sending the interference signals I5, I6 and I1 into the lock-in amplifiers. The out-of-plane displacement in the Z-direction can be obtained by sending the interference signals I7 and I1 into the lock-in amplifier. Similarly, in detection part III (framed by the orange dashed line), the interference signals I8, I9 and I10 measured by detectors D8, D9 and D10 can be written as

I8cos[ωt+(ϕX0ШϕX+1Ш)]I9cos[ωt+(ϕY0ШϕY1Ш)]I10cos[ωt+k(lrШlmШ)]

The in-plane and out-of-plane displacements can similarly be obtained by using detection part III. Furthermore, the three rotation angles of θX, θY, and θZ can be acquired by comparing the measurement results between the corresponding detection parts. For example, the θX angle can be obtained by comparing the measurement results of the out-of-plane displacement measured by detection parts I and II. The variation of the ∆θX angle is given by

ΔθX=tan1(ΔlZIΔlZIISY),
where ∆lZI and ∆lZII represent out-of plane displacements obtained by detection parts I and II, and SY stands for the distance between the two focused light beams in detection parts I and II along the Y axis. In contrast, the rotation angle of θY can be acquired by comparing the measurement results of out-of-plane displacement measured by detection parts II and III. The variation of the rotation angle of ∆θY is formulated as follows:
ΔθY=tan1(ΔlZIIΔlZШSX),
where ∆lZII and ∆lZIII are the out-of plane displacement obtained by detection parts II and III and SX means the distance between the two focused light beams in detection parts II and III along the X axis. Moreover, the rotation angle of θZ can be achieved by comparing the measurement results of in-plane displacement on the Y-axis measured by detection parts I and II. The variation of the rotation angle of ∆θZ can be expressed as
ΔθZ=tan1(ΔlYIΔlYIISY),
where ∆lYI and ∆lYII indicate the in-plane displacement obtained by detection parts I and II and SY is the distance between the two focused light beams in detection parts I and II along the Y axis. As can be seen, the 3-DOF rotation angle (θX, θY, θZ) information can be easily obtained by using the proposed optical configuration.

3. System performance test

3.1 Experimental setup

The experimental configuration of our 6-DOF laser interferometer is shown in Fig. 3. A He-Ne laser (632.8 nm) with 45° polarization direction was used as a light source. A sawtooth wave signal with a half wave voltage was generated by a function generator (Tektronix, model: AFG3022B) and passed into the EOM (Newport, model: 4002) for heterodyne modulation. The difference in frequency between the p- and s-polarizations of the heterodyne light source was 16 kHz. The beam dividing method was used to achieve detection configurations I, II, and III by sharing the same light source. It is worth mentioning that the light intensity was evenly separated by the special beam splitter (Thorblabs, model: BS022, 70:30 (R:T), a 1 inch cube) placed after the beam expander. A 2D grating with a 2.8 μm grating pitch was fabricated by holography then mounted on a 6-DOF positioning stage (Physik Instrumente, model: PI-P-562.6CD). Three suitable focusing lenses with focal lengths of f = 20 mm were chosen to focus the beams on the grating. The lens characteristics were chosen such that the diffraction orders would overlap. Three groups of HWPs were placed before the grating for the three detections, each group containing two special half HWPs which were placed perpendicular to the fast axis at 45° and 145°. Seven photo-detectors (Thorlabs, model: PDA-36A) were used to acquire the interference signals. A software lock-in amplifier programed by using graphical language (National Instruments, Labview) was developed and utilized to calculate the phase difference between the corresponding interference signals in the corresponding direction. To demonstrate the facility and performance of our proposed interferometer, several experiments for 3-DOF displacement (X, Y, Z), 3-DOF rotation angle (θX, θY, θZ), repeatability and measurement speed limitation, were performed.

 

Fig. 3 Experimental set-up for 6-DOF measurement.

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3.2 Measurement results of 6-DOF displacement and angle tests (X, Y, Z, θX, θY, θZ)

To demonstrate that the proposed laser interferometer was capable of measuring the 6-DOF displacement and angle information, the positioning stage was operated in a closed-loop configuration and asked to move along the X, Y, Z, θX, θY, and θZ directions, respectively. An internal capacitance sensor was used to simultaneously measure the movement of the stage by comparing the measurement results obtained by our proposed interferometer. The positioning stage was driven by two different waveforms and amplitudes. Figure 4 shows the 3-DOF displacement data experimentally obtained from the step wave (amplitudes 20 nm) and random signal tests. The difference between these and the 3-DOF displacement measurement results obtained using the commercial capacitance sensor (Capacitance sensor, black, straight line) and our method (interferometer: red (X), blue (Y), green (Z), line denoted with circles) can clearly be seen. The experimental results give almost the same displacements in the X, Y, and Z directions regardless of whatever a periodic signal or a random signal is used. It is worth noting here that the curves obtained using our proposed laser interferometer design are as linear as those obtained with the internal capacitance sensor. This demonstrates that our proposed interferometer has the ability and potential to measure 3-DOF displacement with a resolution comparable to that of a commercial sensor.

 

Fig. 4 Measurement results for 3-DOF displacement on the X-, Y-, and Z-axes.

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In addition, our method was used to measure 3-DOF rotation angles in the θX, θY, and θZ directions. According to the measurement principle, the three rotation angles (θX, θY, θZ) are obtained by comparing the measurement results in each corresponding detection part. A calibration procedure is carried out to eliminate the coupling effect between the measurement results for these three rotation angles by comparing measurements made with a precise 6-DOF positioning stage obtained before doing the experiments. The stage was made to move following triangular and random signal waveforms, with amplitudes of about 40 μrad and 10 μrad, respectively. The measurement results obtained with the proposed method and the internal capacitance sensor are shown in Fig. 5. Clearly, it can be seen that the curves show almost the same measurement results and behaviors. It is worth noting that the measurement curves obtained using our method (line denoted with circles) are as linear as those obtained using the capacitance sensor (solid line). Obviously, this demonstrates that the proposed laser interferometer is as capable of measuring 3-DOF rotation angles as a commercial sensor. These above results demonstrate that our interferometer has the ability to measure 6-DOF displacement and rotation angles simultaneously without reorganizing the optical setup.

 

Fig. 5 Measurement result for 3-DOF rotation angle in the θX-, θY-, and θZ-directions.

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3.2 Repeatability tests

In this next set of experiments for testing system repeatability, the positioning stage was driven to produce 75 nm of displacement and 2.5 μrad rotation angle using steps of 15 nm and 0.5 μrad, respectively. In addition, the internal capacitance sensor was used to simultaneously measure the movement of the stage for comparison with the measurement results obtained using our method. The experimental results are shown in Fig. 6. As can be seen in the figure, the trend and behavior of the displacement and rotation angle measured by our method conform to those measured by the capacitance sensor. Clearly, the experimental results obtained by our method and the capacitance sensor are in agreement with each other along each axis. Moreover, the experimental results show that the repeatability of our method is better than 2 nm in the X, Y, and Z dimensions and 0.1 μrad in the θX, θY, and θZ directions. The above measurement results demonstrate that our proposed interferometer has high repeatability on each corresponding axis.

 

Fig. 6 Measurement results for 6-DOF repeatability.

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3.3 Measurement speed limitation tests

Another important feature of any displacement measurement technique is the measurement speed. To test the maximum measurement speed of our proposed method, a precision double layer XY stepper (model: XYS-50; Measure control, Inc.) was used to move the stage over a 10 mm traveling range with four motion speeds (200 μm/s, 400 μm/s, 600 μm/s, and 800 μm/s). The internal linear encoder was used to simultaneously measure the stage movement for comparison with the measurement results obtained by our proposed interferometer. The measurement results obtained by the linear encoder and our interferometer are shown in Fig. 6. As shown in Fig. 7, it can be seen that the trend and behavior of the displacement measured by our method is conformed to the displacement measured by the linear encoder except for the 800 μm/s. As can be seen from the local enlarged area, the measurement results obtained with our method (red rectangles) are smaller than the results acquired by the linear encoder (solid black line) which is due to the problem of phase unwrapping. As far as the phase unwrapping is concerned, the performance of our laser interferometer is limited by the calculation speed of the software lock-in amplifier. The maximum speed we can achieve for each measurement axis is about 600 μm/s. For much higher speed tests (e.g., 1 m/s), a phase meter circuit can be incorporated. This can be a goal of future work. However, it is important here to stress that the current achieved measurement speed is compatible with large-scale near-field probing already demonstrated.

 

Fig. 7 Measurement results for the speed limitation tests.

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3.4 Stability tests

To demonstrate that our proposed interferometer has high system stability, the stage was held stationary and the phase variation over 5 minutes was measured. The system stability could be accurately estimated from this. Figure 8 shows the results for all 6-DOF directions. As shown by the six curves, the stability of the proposed method is actually better than about 20 nm in the X, Y, and Z directions and about 600 nrad in the θX, θY, and θZ directions, without any compensation. Moreover, according to the measurement principle for θY given in Eq. (15), the value of θY can be acquired by comparing the measurement results of OP displacement obtained for detection parts II and III. As can be seen in Fig. 2, the location of the set-up of detection part III is higher than that of detection parts I and II. This means that the measurement results obtained by detection part III are more easily influenced by environmental disturbances. Therefore, the stability for θY is not such high as the other rotation angles.

 

Fig. 8 Measurement results for stability tests.

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In addition to the long term stability, the results clearly illustrate the measurement resolution of our proposed method. The measurement resolution of our interferometer can be defined as the minimum measured amount of displacement and angle. These two values can be obtained along with checking for high frequency noise. The results show that the measurement resolutions of the displacement and angle are better than 2 nm and 0.05 μrad if electrical noise is considered.

3.5 Measurement range and tolerance range tests

The measurement range of IP displacement measurement, which depends on the width of the diffraction grating, is about 50 mm in each detection part. One can use a wider diffraction grating to increase the IP measurement range. Moreover, the measurement range of OP displacement measurement, which depends on the depth of focus of the focusing lens, is about 1.2 mm in our current set-up. One can use a wider depth of focus to increase the OP measurement range. In addition, the measurement ranges for the three rotation angles can be confirmed by a rotation limitation test. In our current set-up, the measurement ranges of θX, θY, and θZ are about 800 μrad, 800 μrad, and 1000 μrad, respectively. One can decrease the distance between the two focused light beams in the corresponding detection parts to increase the measurement ranges of θX, θY, and θZ.

The tolerance ranges given for the interval in which the IP or OP displacements can be measured are confirmed by a simple test. The 6-DOF positioning stage was moved with amplitude of 3.5 μm by sending a continuous sine waveform from a function generator. Then, a XY stepper was used to move the 6-DOF positioning stage along the propagation direction until the displacement can no longer be measured correctly. The tested tolerance ranges were found to be about 1.2 mm for IP and OP displacement measurements.

4. Discussion

The proposed interferometer consists of many components and alignment is a big concern. In general, error resulting from misalignment is difficult to avoid and can be seen as an uncertainty. Actually, we think there are two dominant sources of misalignment error. One comes from the misalignment of polarization components and has already discussed in our previous work [20]. The diffraction grating plays a very important role in our proposed interferometer, meaning that misalignment of the assembly between the grating and the positioning stage is the other dominant source of error and should be taken into account. To test for this problem we consider three misalignment angles arising from the yaw (θX), pitch (θY), and roll (θZ), as shown in Fig. 9. The results are discussed in detail below.

 

Fig. 9 Sketch of the misalignment caused by the angles of yaw, pitch, and roll.

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Case 1: yaw angle (θX) occurs

First consider the effect of a misalignment of the yaw angle on 3-DOF displacement measurement. Actually, only IP detection on the Y-axis and OP detection on the Z-axis will be influenced by the occurrence of yaw. According to the measurement principle for displacement on the Y-axis, the amount of influence (ΔdeY) caused by the yaw factor can be given by

ΔdeY=ΔΦY2π(ppyaw)=ΔdY(1cosθX),
where θX stands for the yaw angle caused by the misalignment, and pyaw indicates the pitch size with the misalignment of the yaw angle. Since the yaw angle causes a change in the spacing between the interference fringes, the contrast of the interference signal will be affected. However, this phenomenon will not induce any measurement error on the Z-axis. Therefore, in this case, the amount of influence (ΔdeZ) caused by the yaw factor on the Z-axis can be ignored.

Furthermore, the occurrence of yaw angle misalignment during 3-DOF rotation angle measurement will not influence the measurement results for θX and θY, but will affect the measurement result for θZ. According to the measurement principle for the rotation angle of θZ, the amount of influence (ΔθeθZ) caused by the misalignment of the yaw angle can be written as

ΔθeθZ=ΔθZΔθZ'=tan1(ΔlYIΔlYIISY)tan1((ΔlYIΔlYII)cosθXSY),
where ΔθZ′ and ΔθZ indicate the rotation angle of θZ obtained with and without misalignment of the yaw angle, and θX stands for the yaw angle caused by the misalignment.

Case 2: pitch angle (θY) occurs

Moreover, next let us consider the occurrence of pitch angle situation in 3-DOF displacement measurement. In fact, the situation is similar to the case mentioned above when yaw occurs. With the exception of IP detection on the Y-axis, the measurement results indicate that IP detection on the X-axis and OP detection on the Z-axis will be affected when pitch occurs. The amount of measurement error (ΔdeX) on the X-axis caused by the pitch factor can be formulated as follows:

ΔdeX=ΔΦX2π(pppitch)=ΔdX(1cosθY),
where θY represents the misalignment of the pitch angle, and ppitch is the pitch size with the misalignment of the pitch angle.

The influence (ΔdeZ) caused by the pitch factor on the Z-axis can be ignored for the same reason as mentioned in case 1. The measurement results obtained with our device for these three rotation angles will not be influenced by the effect of pitch.

Case 3: roll angle (θZ) occurs

In the case of the roll angle, the measurement results for IP detection on both the X- and Y-axes will be affected, but OP detection on the Z-axis will not as there is no change in optical path difference between the reference and measurement beams. Since the 2D grating used in our system has the same pitch size in both X- and Y- directions, the amount of measurement errors (ΔdeX) on the X-axis and (ΔdeY) Y-axis caused by the roll factor should be the same and can be formulated as

ΔdeX=ΔdeY=ΔΦq2π(pproll)=ΔΦq2πp(11cosθZ)=Δdq(11cosθZ),
where θZ represents the misalignment of the roll angle, proll means the pitch size with the misalignment of the roll angle, and (Δdq) stands for the displacement measurement result in q (q = X or Y) direction.

Furthermore, in the case of 3-DOF rotation angle measurement, the occurrence of misalignment of the roll angle will not affect the measurement results for θX and θY since there is no optical path difference change between reference and measurement beams on the Z-axis. But, the measurement result for θZ will be affected when roll occurs. According to the measurement principle for the rotation angle of θZ, the amount of influence (ΔθZ) caused by the pitch factor can be given by

ΔθeθZ=ΔθZΔθZ'=tan1(ΔlYIΔlYIISY)tan1[((ΔlYIΔlYII)SY1cosθZ)],
where ΔθZ′ and ΔθZ mean the rotation angle of θZ obtained with and without the misalignment of the roll angle, and θZ stands for the roll angle caused by the misalignment.

Usually, the amount of the influence resulting from the misalignment is estimated by using the experimental specifications and conditions. According to the experimental specifications and conditions in our current set-up, if the three misalignment factors (θX, θY and θZ) are controlled within 0.05 degrees while the stage is made to move displacements of 10 mm, 5mm and 1 mm accordingly along the X-axis, Y-axis and Z-axis, measurement errors of 7.62 nm on the X- axis and 3.81 nm on the Y-axis will occur.

5. Conclusion

A grating-based interferometer for 6-DOF displacement and angle measurement is proposed in this study. This technique has the advantages of heterodyne interferometry, grating-shearing interferometry, and Michelson interferometry. The 2-DOF in-plane displacement can be acquired by measuring the phase shift of the interference signals from the moving grating. In addition, the out-of-plane displacement can be acquired by detecting the optical-path difference between the reference beam and the reflection beam from the semi-transmission grating. Three detection parts are arranged using the beam dividing method so that the 3-DOF displacements (X, Y, Z) and 3-DOF angle information (θX, θY, θZ) can be measured simultaneously.

Both the feasibility and performance of the proposed interferometer have been addressed and demonstrated in several experiments. The experimental results show that our proposed interferometer is capable of measuring 6-DOF displacements and angle information without changing the optical configuration. The actual measurement resolutions of the displacement and angle are about 2 nm and 0.05 μrad, respectively. Compared with other commercial measurement instructions, this laser interferometer has the advantages of high resolution, high stability, relatively straightforward operation, and high flexibility.

Acknowledgments

This study was supported by the Ministry of Science and Technology, Taiwan, under contract MOST 103-2221-E-011-080 and NSC-102-2221-E-011-092. The authors cordially thank Prof. J-Y Lee (National Central University, Taiwan) for his assistance.

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9. K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009). [CrossRef]  

10. K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012). [CrossRef]  

11. K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009). [CrossRef]  

12. O. Sasaki, C. Togashi, and T. Suzuki, “Two-dimensional rotation angle measurement using a sinusoidal phase-modulating laser diode interferometer,” Opt. Eng. 42(4), 1132–1136 (2003). [CrossRef]  

13. A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012). [CrossRef]  

14. J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006). [CrossRef]  

15. F. Qibo, Z. Bin, C. Cunxing, K. Cuifang, Z. Yusheng, and Y. Fenglin, “Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide,” Opt. Express 21(22), 25805–25819 (2013). [CrossRef]   [PubMed]  

16. Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005). [CrossRef]  

17. A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010). [CrossRef]  

18. W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011). [CrossRef]  

19. X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013). [CrossRef]  

20. H. L. Hsieh, J. C. Chen, G. Lerondel, and J. Y. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011). [CrossRef]   [PubMed]  

21. D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996). [CrossRef]  

References

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  1. C. M. Wu, “Heterodyne interferometric system with subnanometer accuracy for measurement of straightness,” Appl. Opt. 43(19), 3812–3816 (2004).
    [Crossref] [PubMed]
  2. F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998).
    [Crossref]
  3. W. T. Estler, “High-accuracy displacement interferometry in air,” Appl. Opt. 24(6), 808–815 (1985).
    [Crossref] [PubMed]
  4. M. Nevièvre, E. Popov, B. Bojhkov, L. Tsonev, and S. Tonchev, “High-accuracy translation-rotation encoder with two gratings in a Littrow mount,” Appl. Opt. 38(1), 67–76 (1999).
    [Crossref] [PubMed]
  5. A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992).
    [Crossref]
  6. J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
    [Crossref]
  7. M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
    [Crossref]
  8. S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48(13), 2566–2573 (2009).
    [Crossref] [PubMed]
  9. K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
    [Crossref]
  10. K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
    [Crossref]
  11. K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
    [Crossref]
  12. O. Sasaki, C. Togashi, and T. Suzuki, “Two-dimensional rotation angle measurement using a sinusoidal phase-modulating laser diode interferometer,” Opt. Eng. 42(4), 1132–1136 (2003).
    [Crossref]
  13. A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
    [Crossref]
  14. J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
    [Crossref]
  15. F. Qibo, Z. Bin, C. Cunxing, K. Cuifang, Z. Yusheng, and Y. Fenglin, “Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide,” Opt. Express 21(22), 25805–25819 (2013).
    [Crossref] [PubMed]
  16. Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
    [Crossref]
  17. A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
    [Crossref]
  18. W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
    [Crossref]
  19. X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
    [Crossref]
  20. H. L. Hsieh, J. C. Chen, G. Lerondel, and J. Y. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011).
    [Crossref] [PubMed]
  21. D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
    [Crossref]

2013 (2)

F. Qibo, Z. Bin, C. Cunxing, K. Cuifang, Z. Yusheng, and Y. Fenglin, “Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide,” Opt. Express 21(22), 25805–25819 (2013).
[Crossref] [PubMed]

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

2012 (2)

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

2011 (3)

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

H. L. Hsieh, J. C. Chen, G. Lerondel, and J. Y. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011).
[Crossref] [PubMed]

2010 (1)

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

2009 (3)

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48(13), 2566–2573 (2009).
[Crossref] [PubMed]

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

2008 (1)

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

2006 (1)

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

2005 (1)

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

2004 (1)

2003 (1)

O. Sasaki, C. Togashi, and T. Suzuki, “Two-dimensional rotation angle measurement using a sinusoidal phase-modulating laser diode interferometer,” Opt. Eng. 42(4), 1132–1136 (2003).
[Crossref]

1999 (1)

1998 (1)

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998).
[Crossref]

1996 (1)

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

1992 (1)

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992).
[Crossref]

1985 (1)

Arai, Y.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Bae, E. D.

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

Bin, Z.

Bojhkov, B.

Chang, R. S.

Chang, W. Y.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Chen, C. D.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

Chen, J. C.

Chen, J. H.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Chen, K.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Chen, K. H.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Chen, Q.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Chen, W. W.

Chen, Y. C.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

Cheng, C. H.

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Chiu, H. S.

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

Chiu, M. H.

S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48(13), 2566–2573 (2009).
[Crossref] [PubMed]

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

Cuifang, K.

Cunxing, C.

Demarest, F. C.

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998).
[Crossref]

Dian, S.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Estler, W. T.

Fenglin, Y.

Gao, W.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Hosono, K.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

Hsieh, H. L.

Ito, S.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Kao, F. H.

Keem, T.

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

Kim, S. W.

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

Kim, W.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

Kimura, A.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Lai, C. W.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

Lee, J. Y.

Lerondel, G.

Li, X.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Lijiang, Z.

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Lin, D.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Lin, J. Y.

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

Lu, S. H.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Muto, H.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

Nevièvre, M.

Oh, J. S.

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

Popov, E.

Qibo, F.

Saito, Y.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

Sasaki, O.

O. Sasaki, C. Togashi, and T. Suzuki, “Two-dimensional rotation angle measurement using a sinusoidal phase-modulating laser diode interferometer,” Opt. Eng. 42(4), 1132–1136 (2003).
[Crossref]

Shi, L.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

Shih, B. Y.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

Shimizu, Y.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

Shyu, L. H.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

Su, D. C.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

Suzuki, T.

O. Sasaki, C. Togashi, and T. Suzuki, “Two-dimensional rotation angle measurement using a sinusoidal phase-modulating laser diode interferometer,” Opt. Eng. 42(4), 1132–1136 (2003).
[Crossref]

Teimel, A.

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992).
[Crossref]

Togashi, C.

O. Sasaki, C. Togashi, and T. Suzuki, “Two-dimensional rotation angle measurement using a sinusoidal phase-modulating laser diode interferometer,” Opt. Eng. 42(4), 1132–1136 (2003).
[Crossref]

Tonchev, S.

Tsonev, L.

Wang, S. F.

Wu, C. C.

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Wu, C. M.

Wu, J.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Wu, T. H.

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

Yan, J.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Yang, T. H.

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Yin, C.

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

Yusheng, Z.

Zeng, L.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

Appl. Opt. (4)

Int. J. Mach. Tools Manuf. (1)

J. S. Oh, E. D. Bae, T. Keem, and S. W. Kim, “Measuring and compensating for 5-DOF parasitic motion errors in translation stages using Twyman-Green interferometry,” Int. J. Mach. Tools Manuf. 46(14), 1748–1752 (2006).
[Crossref]

J. Opt. (1)

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacements,” J. Opt. 27(1), 19–23 (1996).
[Crossref]

Manuf. Technol. (1)

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” Manuf. Technol. 60(1), 515–518 (2011).
[Crossref]

Meas. Sci. Technol. (2)

Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16(10), 2030–2037 (2005).
[Crossref]

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9(7), 1024–1030 (1998).
[Crossref]

Measurement (1)

K. H. Chen, H. S. Chiu, J. H. Chen, and Y. C. Chen, “An alternative method for measuring small displacements with differential phase difference of dual-prism and heterodyne interferometry,” Measurement 45(6), 1510–1514 (2012).
[Crossref]

Opt. Commun. (1)

K. Chen, J. H. Chen, S. H. Lu, W. Y. Chang, and C. C. Wu, “Absolute distance measurement by using modified dual-wavelength heterodyne Michelson interferometer,” Opt. Commun. 282(9), 1837–1840 (2009).
[Crossref]

Opt. Eng. (2)

O. Sasaki, C. Togashi, and T. Suzuki, “Two-dimensional rotation angle measurement using a sinusoidal phase-modulating laser diode interferometer,” Opt. Eng. 42(4), 1132–1136 (2003).
[Crossref]

K. H. Chen, J. H. Chen, C. H. Cheng, and T. H. Yang, “Measurement of small displacements with polarization properties of internal reflection and heterodyne interferometry,” Opt. Eng. 48(4), 043606 (2009).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

J. Y. Lin, K. H. Chen, and J. H. Chen, “Measurement of small displacement based on surface plasmon resonance heterodyne interferometry,” Opt. Lasers Eng. 49(7), 811–815 (2011).
[Crossref]

Precis. Eng. (4)

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14(3), 147–154 (1992).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and Z. Lijiang, “Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Sens, Actuator A-Phys. (1)

M. H. Chiu, B. Y. Shih, C. W. Lai, L. H. Shyu, and T. H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens, Actuator A-Phys. 141(1), 217–223 (2008).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 System configuration for 3-DOF displacement measurement.
Fig. 2
Fig. 2 Optical configuration for 6-DOF measurement.
Fig. 3
Fig. 3 Experimental set-up for 6-DOF measurement.
Fig. 4
Fig. 4 Measurement results for 3-DOF displacement on the X-, Y-, and Z-axes.
Fig. 5
Fig. 5 Measurement result for 3-DOF rotation angle in the θX-, θY-, and θZ-directions.
Fig. 6
Fig. 6 Measurement results for 6-DOF repeatability.
Fig. 7
Fig. 7 Measurement results for the speed limitation tests.
Fig. 8
Fig. 8 Measurement results for stability tests.
Fig. 9
Fig. 9 Sketch of the misalignment caused by the angles of yaw, pitch, and roll.

Equations (21)

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E EOM = e i( kl+ϕ ) ( e iωt/2 e iωt/2 ),
A=D=J( 0 ) E EOM ( e i ωt /2 e i ωt /2 ),B=C=J( 180 ) E EOM ( e i ωt /2 e i ωt /2 ),
ϕ qm =2mπ l q /p,
I 2 | P 2 ( 0 )( B X0 + D X+1 ) | 2 cos[ ωt+( ϕ X0 ϕ X+1 ) ]. I 3 | P 3 ( 0 )( B Y0 + A Y1 ) | 2 cos[ ωt+( ϕ Y0 ϕ Y1 ) ]
I 1 cos(ωt).
Δ Φ q =2πΔ l q /p,
Δ l q = pΔ Φ q / 2π .
E r = e i( k l r + ϕ r ) ( e iωt/2 e iωt/2 ), E m = e i( k l m + ϕ m ) ( e iωt/2 e iωt/2 ),
I 4 = | P 4 ( 45 )( E r + E m ) | 2 cos[ ωt+k( l r l m ) ].
Δ Φ Z =k( l r l m )=2Δ l Z .
Δ l Z = Δ Φ Z 2k = Δ Φ Z λ 4π ,
I 5 cos[ ωt+( ϕ X0II ϕ X+1II ) ] I 6 cos[ ωt+( ϕ Y0II ϕ Y1II ) ] I 7 cos[ ωt+k( l rII l mII ) ]
I 8 cos[ ωt+( ϕ X0Ш ϕ X+1Ш ) ] I 9 cos[ ωt+( ϕ Y0Ш ϕ Y1Ш ) ] I 10 cos[ ωt+k( l rШ l mШ ) ]
Δ θ X = tan 1 ( Δ l ZI Δ l ZII S Y ),
Δ θ Y = tan 1 ( Δ l ZII Δ l ZШ S X ),
Δ θ Z = tan 1 ( Δ l YI Δ l YII S Y ),
Δ d eY = Δ Φ Y 2π (p p yaw )=Δ d Y ( 1cos θ X ),
Δ θ eθZ =Δ θ Z Δ θ Z ' = tan 1 ( Δ l YI Δ l YII S Y ) tan 1 ( ( Δ l YI Δ l YII )cos θ X S Y ),
Δ d eX = Δ Φ X 2π (p p pitch )=Δ d X ( 1cos θ Y ),
Δ d eX =Δ d eY = Δ Φ q 2π (p p roll )= Δ Φ q 2π p( 1 1 cos θ Z )=Δ d q ( 1 1 cos θ Z ),
Δ θ eθZ =Δ θ Z Δ θ Z ' = tan 1 ( Δ l YI Δ l YII S Y ) tan 1 [ ( (Δ l YI Δ l YII ) S Y 1 cos θ Z ) ],

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