## Abstract

A novel angular probe using the Fabry-Perot etalon and angular scanning technique is proposed for absolute angular displacement determinations in this paper. The measurement theory is first derived, a setup constructed to implement the angular probe is then introduced and analyzed, and the experimental results from the uses of the setup are finally presented. The setup analyses reveal that the probe is with high measurement resolution and sensitivity. The experimental results not only confirm the validity, stability, accuracy, and repeatability, but also show an application of the angular probe.

©2010 Optical Society of America

## 1. Introduction

Many instruments are capable of examining angular errors (pitch and yaw) of precision translation stages: the autocollimators [1,2], schemes using total internal reflection effect [3–11] or excitation of surface plasma waves [12,13], and heterodyne angular interferometers [14,15]. Among them, the interferometers are with the advantages over the others of high measurement resolution and sensitivity. However, in order to eliminate directional ambiguity, they execute the inspections by extracting the phase difference of two interference signals. The measurements are thus referred to incremental detections, any noise caused by environmental fluctuations may create measurement errors that persist even when the fluctuations disappear, and any light beam interruption during the test destroys the testing with no recovery.

The authors have developed angular interferometers based on using low-coherent light source [16,17]. They not only keep the inherent performance of an interferometer, i.e. high measurement resolution and sensitivity, but also possess the advantage of absolute angular displacement determinations. However, they are with the drawbacks of occupying large space volumes and using expensive components. A novel angular probe based on using the Fabry-Perot etalon and angular scanning technique is thus proposed and introduced in this paper.

## 2. The angular probe

A Fabry-Perot (F-P) etalon is a transparent plate with two reflecting surfaces, in which an incident beam is split into multiple reflected and transmitted beams. As that shown in Fig. 1
, once the transmitted beams are guided to interfere with each other, the interference intensity, **I _{t}**, as well as its maximum,

**I**, has a relation [18] of

_{tmax}**A(θ)**is known as the Airy function,

**F**is the coefficient of finesse of the etalon, and

**ϕ**is the phase difference between successive transmitted beams. Let

**λ**denote the beam wavelength,

**d**and

**n**represent the thickness and refractive index of the etalon, respectively,

**ϕ**can be expressed asWhere

**θ**, the beam angle inside the etalon, and

_{t}**θ**, the angle of the incident beam, are governed by the Snell’s law, i.e.

Equations (2) and (3) imply that **ϕ** decreases as **θ** increases. And Eq. (1) indicates $0<A(\theta )\le 1$ and **A(θ)** has sharp spikes if **F** is large. These sharp spikes inspire us that the Fabry-Perot etalon can be employed for precise angular measurements. This will be exploited in the manner described below.

The schematic diagram of the proposed angular probe is presented in Fig. 2(a)
. The first polarizing beam-splitter (PBS_{1}) separates the linearly polarized beam from a laser source into two linearly polarized sub-beams, L_{1} and L_{2}. These two sub-beams are equal in amplitude, they travel through the Fabry-Perot etalon and then arrive to the second polarizing beam-splitter (PBS_{2}). The PBS_{2} finally directs L_{1} and L_{2} to impinge on the photo-detectors PD_{1} and PD_{2}, respectively. It is noticed that PBS_{1} and PBS_{2} can be replaced by non-polarizing ones, however, stoppers should be carefully placed in front of photo-detectors PD_{1} and PD_{2} in order to block the transmitted and reflected parts of L_{2} and L_{1}, respectively.

The etalon and PBS_{1}are aligned so the incident angles of L_{1} and L_{2} are

**θ**is the angle corresponding to a spike maximum,

_{c}**γ**is the width of the middle-height of the spike, and

**Δθ**is the rotation angle of the etalon. If

**Δθ**is small, the Taylor expansion theory gives

**A**and

_{1}**A**are the Airy functions of L

_{2}_{1}and L

_{2}, respectively. And, by using the chain rule and Eqs. (1)~(3), we get

**A(θ)**=1/2, and it infers that $1+F{\mathrm{sin}}^{2}(\varphi /2)=2$ and$\mathrm{sin}(\varphi /2)\mathrm{cos}(\varphi /2)=\mp \sqrt{1/F}$. Let us further assume that

**θ**is small, then

_{c}**I**,

_{tmax}=I_{t1}+I_{t2}**I**and

_{t1}**I**are the intensities of L

_{t2}_{1}and L

_{2}, respectively,

**S**is a non-dimensional intensity, andWhich,

**K**, is defined as the sensitivity of the angular probe.

Note the etalon is placed on a rotation stage driven by a piezo-actuator. As the etalon module, the assembly of the etalon and the rotation stage, is rotated, one can command the rotation stage to bring the etalon to experience an angular scanning and then find the return angle corresponding to **S**=0. Since the etalon, according to Eq. (12), is at the position of **Δθ**=0 while the stage is at the return angle. The rotation angle of the etalon module can thus be determined, and it is equal to the inverse of the return angle. Also note that the measured angles are based on the datum corresponding to **Δθ**=0, the measurements belong to the category of absolute detections.

## 3. Experimental setup

An angular probe as shown in Fig. 2(a) was installed. Besides, a control and signal processing system, as that shown in Fig. 2(b), adopted for realizing the measurements of the angular probe was also constructed.

The control and signal processing system consists of a personal computer and stage controller/driver. The computer grabs the intensity signals, **I _{t1}** and

**I**, delivered from the photo-detectors PD

_{t2}_{1}and PD

_{2}, communicates with the rotation stage via the use of the stage controller/driver, and executes the programs of scanning, receiver, calculator and display. The stage controller/driver is NC5111-C from Nano Control Co.

The scanning commands the computer to deliver an analog signal to drive the stage to undergo an angular scanning after the etalon module is moved by a test object. The receiver informs the computer to simultaneously record the beam intensities and the rotation angles of the stage while the stage is scanning. The calculator calculates the non-dimensional intensities, **S**, and extracts the angle (i.e., the return angle) with respect to **S**=0. And the display exhibits the rotation angles of the etalon module on the monitor.

The angular probe is with a He-Ne laser source (**λ**=632.8nm) and an etalon module as that shown in Fig. 3
. In which, the etalon has a thickness of **d**=10mm, refractive index of **n**=1.46, coefficient of finesse of **F**=200, and surface flatness within λ/10. The rotation stage is NS5311-C from Nano Control Co.

The angular positioning resolution of the rotation stage is 0.01 arc-sec. It is also the smallest angular change the angular probe can detect, the angular probe thus has a measurement resolution of 0.01 arc-sec.

The etalon was put at **θ _{c}**=2.531deg. The measured Airy function of beam L

_{1}is represented in Fig. 4 (the Airy function of beam L

_{2}is the same as that of L

_{1}but with a shift to the right), it was obtained from the use of the data recorded by the receiver while the etalon was angularly scanned from −315 to 285 arc-sec. From the Airy function shown in Fig. 4, one can see that the spikes away from the one corresponding to

**θ**are with

_{c}**Δθ**larger than 200 arc-sec. The angular probe is thus available while the rotation angle is within 200 arc-sec, since, in this range, only one position makes

**A**and

_{1}**A**be close to 1/2 and

_{2}**S**=0.

From the recorded data, the non-dimensional intensity **S** with respect to small rotation angle **Δθ** was also determined and is presented in Fig. 5
. The least squares line of the S-curve shows that the angular probe is with a sensitivity of **K**=19151. It implies, an arc-sec increment of the rotation angle makes the non-dimensional intensity have a variation of 9.3%.

## 4. Experimental results

The experimental setup of Fig. 2 was used to investigate the stability of the angular probe. The etalon module was put on a rotation stage (103H548-0410 from Sanyo Denki LTD.) driven by a stepping motor, and the probe examined the angular movement of the stage while it was in stationary state. Figure 6 depicts the result of one hour test and shows that there was an angular variation up to 0.35 arc-sec. This variation, defined as the stability of the probe, was induced by environmental noises (response deviations of photo-detectors, vibration, thermal drift, etc.). Since the probe would detect the angles with the compositions due to the noises, the angular probe may inspect an angle with an error up to the amount of the stability.

The setup was then conducted to examine the validity of the angular probe. The rotation stage used in the stability test was calibrated by a heterodyne angular interferometer [15]. It was then moved step by step, at each step the angular movement was determined by the angular probe for ten times. Figure 7 presents the rotation angles and measurement averages at the steps. As indicated in this figure, the consistency between the measured and rotation angles validates the angular probe, the maximum difference, i.e. the difference at step 9, demonstrates the probe has an accuracy of 164.65-163.20=1.45 arc-sec.

Figure 7 also presents the standard deviations of the measurements at the steps. The maximum of the standard deviations, i.e. the deviation at step 8, implies the angular probe has a repeatability of 0.24 arc-sec.

The installed setup was finally utilized to verify the applicability of the angular probe. A translation stage whose moving axis was aligned along the direction of the X-axis was driven 10 times from 0 to 20 cm in 1cm steps. The etalon module of the probe was placed on this stage, and the probe was employed for examining its angular error, i.e. yaw, rotating about the Z-axis [15]. Figure 8 depicts the averages and deviations of the measurements at the steps. Note the deviations, denoted by the error bars, represent the instability of the stage. They are due to the clearance between the saddle and guide of the stage. The capacity to measure the angular error and instability in the stage demonstrates the applicability of the angular probe.

## 5. Discussions

According to Eqs. (1)~(3), the theoretical Airy function of beam L_{1} was calculated and is also presented in Fig. 4. It is obvious that the spikes of the theoretical one are sharper than those of the measured one, consequently the theoretical sensitivity (42486, which was obtained from the use of Eq. (13)) is higher than the probe sensitivity. The flatness error of the etalon surfaces makes the phases of the transmitted beams of L_{1} (and L_{2}) be randomly and slightly deviated from each other, the resonance of the transmitted beams is somewhat disturbed, and the spikes of the measured Airy function are thus blunted. An etalon with higher flatness would sharpen the spikes, it is however expensive in price.

As the probe is used for an angular examination, the computer commands the rotation stage of the etalon module to have an angular scanning, records the intensities corresponding to the scanning angles, and determines the rotation angle by discovering the return angle with respect to **S**=0. This is simple but time consuming. An algorithm, which draws the etalon back to **Δθ**=0 right after the etalon module is moved, is developed to overcome the drawback by the authors successfully. Where the drawing direction and magnitude are based on the detected **S**.

A translation stage is, in general, also with the angular error, pitch, rotating about the Y-axis. Once the PBS_{1} and etalon module are rotated (from the situation shown in Fig. 2(a)) about the X-axis by 90 deg., the angular probe is available for pitch-measurements. Note the pitch produces negligible errors while the probe is utilized for yaw-measurements, and vice versa.

## 6. Conclusions

In summary, an angular probe based on the use of the Fabry-Perot etalon and angular scanning technique was introduced for measuring absolute angular displacements. The theory of the probe was demonstrated, a setup, which is available while the angular displacement is within 200 arc-sec, with a measurement resolution and sensitivity of 0.01 arc-sec and 19151, respectively, was presented. The experimental results from the applications of the setup validate the probe and demonstrate a measurement stability, accuracy, and repeatability of 0.35, 1.45, and 0.24 arc-sec, respectively. These results also present an application demonstrating the use of the angular probe.

## Acknowledgments

The support of the National Science Council, ROC, under grant NSC 97-2221-E-027-011 is gratefully acknowledged.

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