## Abstract

A high resolution roll measurement heterodyne interferometer with differential configurations is proposed in this paper. The proposed interferometer is composed of a pair of measurement beams providing environmental noise immunity. The structure is designed and the mathematic model based on Jones’ matrix for measuring the roll angle is established. Sensitivity is enhanced dramatically because of the opposite phase shift directions of two signals. An experimental setup is built and an amplification factor of 270 is obtained. Correspondingly, a roll angle resolution of 0.13 arcsec is attained using a phase meter with detecting resolution of 0.01°.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The demand for precision translation stage is expanding for its need in rapid development of semiconductor, advanced manufacturing and other researches. Six geometric errors occur along the movement of the translation stage including the position error, straightness errors along two axes perpendicular to the moving direction and three angular errors around three axes corresponding to pitch, yaw and roll. A lot of work has been reported for single degree of freedom (DOF) measurement [1–4] and simultaneously measuring multiple DOFs [5–16]. Heterodyne interferometer is the preferred method to provide most DOF measurements due to its high accuracy and environmental immunity. However, the roll angle, unlike pitch and yaw, is more difficult to measure as an in-plane displacement. It does not bring optical path change in Michelson interferometer nor does it lead to the deflection of the reflected beam in Autocollimator.

From the previous research, the optical method for examining the roll error can be divided into three categories. Firstly, collimated laser position method [13,17–19], using reflectors, transforms roll to the movement of two light spots on a two-dimensional photodetector such as quadrant detector or CCD. Then the roll error can be obtained from the movements of spots and the initial separation distance between the two beams. The second method is optical path difference detection based on heterodyne interferometer [20–22]. It sets a bi-wedge prism and a bi-wedge mirror to replace the target mirror in the plane mirror interferometer. When the bi-wedge prism rolls, the optical path of the measurement arm is changed due to the refractive index difference between the prism and air. Additionally, the designed four centrosymmetric beams increase the stability of the measurement. The last method is also a heterodyne interferometer but based on polarization direction [23–28]. In this method, a beam passes through the transparent plate actuated by the roll rotation and its polarization direction is modulated correspondingly, so the roll angle is conversed into an optical phase shift after interference. Liu [24] and Wu [25] both presented optical layouts based on this method. They used a HWP which is passed through twice and a wave-retardation plate of 178° as the sensor plate, respectively. In our previous work [28], we have demonstrated a simple high resolution configuration with an optimized tilt folding mirror. The above polarization direction methods are all single-path interferometer composed of a reference arm and a measurement arm. However, during the measurement of long distance translation stage, the environmental disturbance can give rise to significant phase difference fluctuations. This is due to the unequal optical path between the reference and measurement beams of the single-path roll measurement interferometer.

In this research, we propose a differential heterodyne interferometer for the measurement of the translation stage roll angle. This interferometer utilizes the previously unused reference beam as another measurement beam passing through the sensor plate. The two interference measurement signals show opposite phase shift direction, which doubles the measurement resolution. Besides, the differential structure minimizes the environmental errors and stabilizes the roll measurement result.

## 2. Measurement principle

#### 2.1 Single-path roll measurement heterodyne interferometer

A typical single-path roll measurement heterodyne interferometer is shown as Fig. 1. An orthogonally polarized dual-frequency laser with frequency *f*_{1} and *f*_{2} are divided into a reference beam and a measurement beam through a beam splitter (BS). The measurement beam proceeds to a quarter wave plate (QWP) and then passes the half wave plate (HWP) which is fixed on the translation stage as the sensor element. Coming back after reflected by an assembly mirror (AM), the beam passes through the HWP again. The reference and measurement beams respectively get interference after passing through the polarizer (P_{A} and P_{B}) and then arrive at the photo detector (PD_{A} and PD_{B}). A phase shift of measurement signal caused by the roll of the HWP is detected by a phase meter from two PDs. The relationship between the phase difference *ψ* and the roll angle *α* is given from Ref [25]. as

*θ*is the parameter representing the azimuth of QWP.

It is known that the turbulence of the atmospheric parameters such as the temperature, pressure, humidity and the CO_{2} content affect the instability of refractive index of air, which is one of the major causes of uncertainties in heterodyne interferometer.

As shown in Fig. 1, we assume that the two detected signals are

*k*

_{A}and

*k*

_{B}are parameters related to the light intensity.

*φ*

_{A1}and

*φ*

_{A2}are the optical phase of two modes with different frequencies

*f*

_{1}and

*f*arriving at the photodetector in Path A.

_{2}*φ*

_{B1}and

*φ*

_{B2}are detected at the photodetector in Path B. We have

*φ*

_{A1}= 2π

*f*

_{1}

*L*

_{A}n/c,

*φ*

_{A2}= 2π

*f*

_{2}L_{A}n/c,

*φ*

_{B1}= 2π

*f*

_{1}

*L*

_{B}n/c and

*φ*

_{B2}= 2π

*f*

_{2}

*L*

_{B}n/c, where

*L*

_{A}and

*L*

_{B}are the optical paths of the reference and measurement arms, respectively.

*n*is the refractive index of air and

*c*is the speed of light in vacuum.

The phase difference of the two signals is calculated as follows

The phase difference is proportional to the refractive index of air *n* and is related to the imbalance between two optical arms and the frequency offset. So, the minor phase difference drift is based on small optical path difference. Hence, we propose a differential structure with approximate balanced arms that not only stabilizes the measurement result but also enhances measurement sensitivity.

#### 2.2 Differential roll measurement heterodyne interferometer

The proposed differential heterodyne interferometer is shown in Fig. 2. A stabilized single-frequency laser is equally divided into two beams by a polarized beam splitter (PBS1) and then diffracted by two acousto-optic modulations (AOMs), respectively. Both positive first order diffraction lights with two different frequencies *f*_{1} and *f*_{2} pass the pinholes (PH1 and PH2) and reach PBS2. Then, the beam with *f*_{1} and the beam with *f*_{2} are merged into one. This beam is further split into two same and parallel parts (Beam 1 and Beam 2) by a beam splitter (BS) and a right angle prism (RAP). Beams 1 and 2 respectively proceed to a quarter wave plate (QWP1 or QWP2) in their own path and subsequently pass a half wave plate (HWP). The HWP is the sensing plate fixed on the translation stage in practice. After folded back by an assembly mirrors (AM), the two beams pass through the HWP again. Then, they are collected separately by two photodetectors (PDs) preceded by two polarizers (P1 and P2). The *x* and *y* axes are set to be parallel to the TM and TE waves of Beam 1 as shown in Fig. 2. The *z* axis is along the propagation direction. The QWPs are oriented at a small angle of *θ*_{1} = *θ* = 2° and *θ*_{2} = *-θ* = −2° to make the measurement beams slightly ellipse polarized. The transparent directions of P1 and P2 are set at the same angles as QWPs in the corresponding path.

The Jones’ matrixes of the input beams (B_{1} and B_{2}), QWP1, QWP2, HWP, P1 and P2 are given as

The matrixes of the received beams are written as

Substituting the above values into Eq. (5), we can get the signals as

The intensity is proportional to the square of the vector’s amplitude. After decoupling the DC term by a high-pass filter, the signals are expressed as follows

From Eq. (8) and Eq. (9), it is observed that the phase shifts of two signals with respect to the same roll angle are in the opposite direction. Then the phase difference between Beams 1 and 2 are given as

The initial phases of the beams (φ_{1} and φ_{2}) is completely irrelevant to the ultimate readout and the phase difference is twice that of the single-path interferometer as compared to Eq. (1), which is also a function of the azimuth *θ* of the QWP and the azimuth *α* of the HWP. The simulations of the phase difference *ψ* versus the roll angle *α* of the differential and single-path roll interferometers for *θ* = 2° are illustrated in Fig. 3. As shown around point A, the local slope is evidently higher than that in other places within that period and the curve is quasi-liner in a limited range. Thus, we define this region as the sensitive area. In this area, the micro roll angle is magnified to observable phase difference. As the moving device rolls, the roll angle is obtained by measuring the phase difference. Providing in the sensitivity area, tan ∆*α* ≈k∆*α* and arctan ∆*α* ≈∆*α*/k, where k is a constant value. The amplification factor can be written as

Note that the amplification factor A of the proposed differential system is twice that of the single-path system, which significantly enhances the measurement resolution. The roll angle in the sensitive area can be calculated as

When *θ* = 2°, the theoretical value of the amplification factor is 228. The amplification factor actually can be improved by decreasing QWP’s azimuth whereas it is at the cost of reducing the light intensity.

#### 2.3 Simple differential roll measurement heterodyne interferometer

A simplified differential heterodyne interferometric configuration with only one QWP and one polarizer (P) for two measurement arms is also proposed as exhibited in Fig. 4. The direction of the coordinate system is the same as Fig. 2. The stabilized single-frequency laser is 45° polarized and equally divided into two beams by a non-polarized beam splitter (NPBS). The diffracted positive first order lights with frequencies *f*_{1} and *f _{2}* are 45° polarized too. Then, the transmitted beam with

*f*

_{1}(TM wave) and the reflected beam with

*f*

_{2}(TE wave) are merged into Beam 1. Similarly, the reflected beam with

*f*

_{1}(TE wave) and transmitted beam with

*f*

_{2}(TM wave) are merged into Beam 2 and further reflected by a RAP to be parallel to Beam 1. Then these two beams proceed to one QWP and pass through HWP (the sensing plate) twice after reflecting by the assembly mirrors (AM). Finally, they are separately collected by two photodetectors (PDs) preceded by a common 45° polarizer (P). The QWP is oriented at a small angle

*θ*= 2° .

The Jones’ matrixes of the input orthogonal polarized beams, QWP, HWP, and P are written as

The matrixes of beams received by PD1 and PD2 are written as

After calculation and DC filtering, the phase difference of two beams is given as same as Eq. (10). Consequently, this simplified differential interferometer with fewer optical plates can achieve high resolution and is easier to build. It should be noted that the roll angle *α,* between the fast axis of HWP and *x* axis, is determined by the roll dimension, not the roll center, so the rotation center of HWP has no influence on the measurement result.

## 3. Experiment and results

To verify the performance of the proposed differential heterodyne interferometer, an experimental setup based on the schematic diagram in Fig. 2 was assembled and shown in Fig. 5. The laser source was a stabilized single-frequency He-Ne laser (05STP910, Melles Griot Co., USA). The frequency shifts of two AOMs (3080-125, Gooch & Housego Co., UK) were set as 50 MHz and 50.05 MHz, respectively, so the heterodyne frequency detected by the photodetector was 50 kHz. The HWP was fixed in a rotatable plate frame and mounted on a tilt stage (Y200GA10, Jiangyun Optoelectronic Technology Co., China) which provided the roll displacement with 0.01° resolution. Two high-speed PIN photodetectors (PAP36A, Thorlabs Co., USA) were used to detect the measurement signals with maximal frequency of 10 MHz. The phase difference was measured by a phase meter (6000A, Clarke-hess Co., USA). A 10% beam splitter was placed behind the PBS2 to direct a beam as a reference so that Beam1 and the reference beam could form a single-path roll interferometer. A commercial interferometer (XL-80, Renishaw Co., UK) with 0.1 arcsec resolution was used to gauge the roll of the tilt table for comparison.

Firstly, we measured the phase differences of the differential and single-path interferometers. The experimental results of phase difference versus roll angle with 2° resolution were shown in Fig. 6. In the roll range of 90°, the result of differential interferometer has four periods, while the result of the single-path interferometer has two periods. The trend of curves matches with the simulations in Fig. 3. It is pretty clear that the sensitive areas of the differential interferometer were located around Points A1, A2, A3 and A4 in Fig. 6. The phase difference in the sensitive area of the differential interferometer was increased twofold comparing with the single-path interferometer.

To calibrate the amplification factor, the HWP was rotated to Point A2 in Fig. 6. Then the tilt stage was rotated from −0.1 to 0.1° at a step of 0.01° along the roll direction. The variation of phase difference is quasi-linear within 0.2° as displayed in Fig. 7. The amplification factor of the line fitted by the least-square method is A = 270, which is a slightly higher than the theoretical value of 228 possibly caused by the smaller azimuth of QWP. Correspondingly, with an interferometer phase meter resolving *δ*ψ = 0.01°, a roll angle resolution of *δ*ψ/A = 0.13 arcsec is attained.

The roll angle of the tilt stage was calculated by dividing the phase difference change by the amplification factor A = 270. A comparison of the rolls obtained by the differential interferometer and the XL-80 interferometer and the residuals between them are shown in Fig. 8. The maximum residual error is about 4 arcsec with a standard deviation of 2.8 arcsec. The residuals were mainly caused by the fluctuation of phase difference due to the atmosphere turbulence and the frequency stability of laser.

The stability of the proposed interferometer was tested for five groups of roll measurements along a 155mm translation stage (LS-110, Physik Instrument Co., Germany). The maximum pitch and yaw angle error of the stage are 15 and 8 arcsec, respectively. The stage is driven at a step of 1mm supporting 10mm/s operating speed. The measurement results for this experiment trail are shown in Fig. 9. The maximum roll error is up to 40 arcsec for the whole travel and the maximum point-to-point standard deviation is about 3.5 arcsec. The exhibited consistent repeatability confirms the validation of the proposed interferometer.

In the roll interferometers, the retroreflector is fixed on the end of the translation stage. Instead, HWP is the sensor plate and moves with the slipper part during the measurement. Since the beams pass through the HWP, there is no Doppler frequency shifts for the received signals. Thus, the proposed roll interferometers are free from Doppler effect and its measurement speed theoretically is not limited by the modulation frequency but only by the calculation speed of phase demodulation. Normally, the roll error of the translation stage is not measured continuously. The HWP moves to a position and pauses for one or two seconds and then moves to the next position. The phase difference is calculated during the pause. Usually, the calculation time is far less than a second. Therefore, we can ignore the limit of the calculation speed of phase demodulation. As for the moving speed between two positions, it is unnecessary to limit it.

## 4. Conclusion

This paper proposed and verified a resolution enhanced heterodyne laser interferometer featured with differential configuration for roll angle measurement. The nearly balanced configuration was sturdy and provided environmental noise immunity. The measurement resolution was enlarged twofold due to the opposite phase shift direction between two measurement signals. To verify the effect of the interferometer, an experimental equipment was set up. The results were compared with the single-path roll measurement interferometer and an enhanced resolution of 0.13 arcsec was confirmed. The feasibility and repeatability were validated by the roll error measurement of a precision translation stage.

## Funding

National Natural Science Foundation of China (NSFC) (61405156).

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