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A displacement sensor combining cavity tuning of a laser with apiezoelectric transducer's subdivision technique for a bidirectionalsampling on the rising and falling flanksZhengqi Zhao, Shulian Zhang, Song Zhang, Yidong Tan, and Yan Li 

Citation: Rev. Sci. Instrum. 82, 115001 (2011); doi: 10.1063/1.3658200 

View online: http://dx.doi.org/10.1063/1.3658200 

View Table of Contents: http://rsi.aip.org/resource/1/RSINAK/v82/i11 

Published by the American Institute of Physics. 

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REVIEW OF SCIENTIFIC INSTRUMENTS 82, 115001 (2011)

A displacement sensor combining cavity tuning of a laser with apiezoelectric transducer’s subdivision technique for a bidirectionalsampling on the rising and falling flanks

Zhengqi Zhao, Shulian Zhang,a) Song Zhang, Yidong Tan, and Yan LiState Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, China

(Received 24 August 2011; accepted 13 October 2011; published online 1 November 2011)A displacement sensor combining cavity tuning of a laser with a piezoelectric transducer’s subdivision

technique is presented. Because of the low gain, the measuring range of the displacement sensor based

on the orthogonally polarized dual-frequency He–Ne laser at 633 nm is limited. The gain coefficient

is proportional to the cube of the wavelength, so the He–Ne laser at 1.15 μm is adopted in research. A

subdivision technique using a piezoelectric transducer enhances the resolution and the bidirectional

sampling of the actuated voltage on its rising and falling flanks effectively amends the hysteresis and

nonlinearity effect. The displacement sensor achieves the resolution of 10 nm in the range of 100 mm.

© 2011 American Institute of Physics. [doi:10.1063/1.3658200]

I. INTRODUCTION

Displacement is one of the most fundamental parameters

in physics. According to different theories, there are two ma-

  jor categories of high-precision displacement measurement:

the electrical method and the optical method. Electrical de-

vices are compact and cheap, but the linear measurement

range is relatively small. Due to the traceability to the wave-

length, applications of laser technology are the most widely

used optical method,1, 2 and a frequency stabilization sys-

tem is used to achieve high precision. The dual-frequency

laser interferometer is a good example because it provides the

large measuring range and a high resolution.3, 4 However, zero

shifts always occur even in laboratory environment, because

the measuring optical path is exposed in the air. Although the

laser frequency stabilization achieves high precision for thewavelength, the measuring system is still sensitive to the air

disturbance.

A solution is to simplify the structure and to avoid

measurements with the optical path exposed to the air. So

we developed a displacement sensor based on cavity tuning

of the orthogonally polarized dual-frequency He–Ne laser at

633 nm.5 The key point of this method is that the laser acts as

a sensor as well as a light source. The structure is simplified

and no optical path is exposed in the air. This displacement

sensor is mainly based on the theories of frequency splitting,

mode competition, and cavity tuning of a laser. The change of 

cavity length, which equals to the displacement of one of the

cavity mirrors, can be calculated by the cavity tuning curves.The displacement sensor achieves the resolution of 79 nm for

a measurement range of 12 mm, with advantages of linearity,

self calibration, and without a frequency stabilization system.

However, the measuring range and stability are restricted

by the gain of the medium. So we chose the He–Ne laser at

1.15 μm considering the higher gain. With the help of a new

a)Author to whom correspondence should be addressed. Electronic mail:zsl-dpi@mail.tsinghua.edu.cn.

signal processing and with piezoelectric transducer’s subdivi-

sion technique, the resolution and the measurement range of 

this improved displacement sensor reaches 10 nm for a mea-surement range of 100 mm. The performance is much better

and all the merits are reserved.

II. MEASUREMENT SYSTEM

The schematic structure of the displacement sensor is

shown in Fig. 1. A half-intracavity laser is composed of a con-

cave output mirror M, a cat’s eye reflector CER, and a He–Ne

laser discharge tube T. All parameters of the discharge tube T

are selected through a series of experiments. W is the window

plate with the anti-reflection coating on both surfaces. Q is

a quartz plate which is obliquely placed to keep a certain an-

gle between the crystal axis and the laser axis. With this setup,

two orthogonal polarized linear beams are generated. Accord-

ing to the crystal optics, we can call them o-light and e-light

separately. The frequency difference can be adjusted by the

rotation of the quartz plate. PZT is a piezoelectric transducer,

and ROD is the rod moving in axial direction in a sleeve. CER,

PZT, and ROD are mounted together. PZT is actuated by a tri-

angular voltage. So CER performs a reciprocating motion in

the axial direction and the cavity length is modulated. PBS is

a Wollaston prism used to separate two orthogonal polarized

linear beams. D1 and D2 are two photoelectrical detectors.

The output consists of two beams from M, separated by PBS

and then projected on D1 and D2.CER is composed of a convex lens and a concave mir-

ror. There are anti-reflection coatings on both surfaces of the

convex lens and a high reflective coating on the left side of 

the concave mirror as shown in Fig. 1. The focal length of 

the convex lens, the radius of curvature of the concave mir-

ror, and the distance between the convex lens and the concave

mirror are all equal. A normal incident paraxial beam will be

reflected back by CER along the entrance way. Even for the

obliquely incident paraxial beam (the tilt angle is small), CER

can still provide high parallelism for the incident and reflected

0034-6748/2011/82(11)/115001/3/$30.00 © 2011 American Institute of Physics82, 115001-1

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115001-2 Zhao et al. Rev. Sci. Instrum. 82, 115001 (2011)

FIG. 1. Schematic structure of displacement sensor.

beam; however, that is impossible for any traditional laser res-

onator mirror. That is the reason why the CER used as a res-

onator mirror can improve the laser stability. If there is a tinyvibration in the process of the displacement measurement, the

application of a CER can reduce the destructive influence.6

III. EXPERIMENTAL METHOD

The reason for using a He–Ne laser at 1.15 μm is

revealed by the laser theory. The active medium of the He–Ne

laser is mainly Doppler broadened, so the line shape function

g D(ν,ν0) is in the form of a Gaussian function. When the

frequency ν equals to the central frequency ν0, g D(ν, ν0) gets

the maximum:7

g Dmax= λ

m

2πk  B T 

1/2

, (1)

where λ is the wavelength, m is the atomic mass, k  B is the

Boltzmann constant, and T is the temperature. When ν equals

to ν0, the gain coefficient G is

G = nυ2 A21

8πc2

m

2πk  B T 

1/2

λ3, (2)

where n is the inverted population density, υ is the ve-

locity of atoms, and A21 is the spontaneous emission rate.

Equation (2) shows that the gain coefficient G is proportional

to λ3. It means that the gain of the laser will be higher for a

longer wavelength for the same active medium. Higher gainpromises better stability and longer displacement measure-

ment range, so the He–Ne laser at 1.15 μm is a better choice

as a displacement sensor.

During the displacement measurement, the PZT is actu-

ated by a high-frequency triangular wave voltage. High pass

filtered voltage signals proportional to the intensities of o-

light and e-light are shown in Fig. 2.

FIG. 2. Voltage signals after high-pass filter.

In Fig. 2, a pulse is generated at the position where the

intensity of o-light equals to that of e-light. The pulses are

used to count the displacement and the pulse equivalent isλ /4.

The sequence of o-light and e-light is opposite when the PZT

moves in the opposite direction. The phenomena can be used

for judgement of the direction by logic algorithm in the digital

circuit. The cavity tuning curves are deviated from the Gauss

shape. Numerical calculation of Lamb theory reveals that the

distortion of the cavity tuning curves is a combined action of self-saturation effects and cross-saturation effects.8, 9 With the

help of CER, the output laser keeps steady when the cavity

length changes by 100 mm. So a measuring range of 100 mm

is achieved.

As it is widely known, there is hysteresis and nonlin-

earity between the actuated voltage and the elongation of a

PZT, and the hysteresis and creep properties of a PZT are

drawbacks.10, 11 The creep of a PZT disappears after several

hundred seconds after powering on, but the hysteresis and

nonlinearity still exist. To weaken the adverse impact in the

range of λ /4, a subdivision method is implemented by the sys-

temic sampling of both flanks of the triangular wave voltages.

As shown in Fig. 2, the laser cavity length is the same at thesystemic sampling positions: position A and position B. The

FIG. 3. Hysteresis and nonlinearity testing of PZT.

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115001-3 Zhao et al. Rev. Sci. Instrum. 82, 115001 (2011)

FIG. 4. Comparing experiments with laser interferometer and micro-moving

platform.

corresponding actuated voltages V A and V B are sampled. V Aand V B should be the same if there is no hysteresis or nonlin-

earity, but in fact they are slightly different from each other.

To make the amendment, the mean value of V A and V B is used

to represent the position A (also the position B). The effec-

tiveness of this method is based on the characteristics of the

PZT. So, the PZT used in the sensor system is tested as shown

in Fig. 3.

The PZT is in an open-loop control and the elon-

gation is tested by microchip a Nd:YAG laser feedback 

interferometer.12

The solid line in Fig. 3 is acquired by cal-culating the mean value of the actuated voltages, which are

sampled at the systemic position where the intensity of o-

light equals to that of e-light on the rising and falling flanks

separately. In the range of λ /4, that is, 288 nm, the nonlinear-

ity is 1.7% and 2.4% when the triangular wave voltages are

on the rising flank and on the falling flank separately. With

the method of bidirectional sampling, the nonlinearity low-

ers to 0.9%. The nonlinearity is significantly improved, and

the slope of the solid line in Fig. 3 is the elongation-voltage

multiplier. By monitoring the change of actuated voltages at

the positions where the intensity of o-light equals to that of 

e-light, subdivision in the range of 288 nm is realized. With

an 8-bit AD converter, the resolution is 1 nm.

IV. EXPERIMENTAL RESULTS

Comparing experiments are implemented in both long

and short range. In the range of 100 mm, the linear mo-tion of the translation stages is measured simultaneously by

the sensor system and a dual-frequency laser interferometer

(Agilent 5529A). In the range of 288 nm, displacement of a

micro-moving platform (PI 762.2L) is measured by the sensor

system. The results are shown in Fig. 4. The linearity in the

range of 100 mm is 7 × 10−6, and the resolution of 10 nm is

realized.

V. CONCLUSIONS

To conclude, a displacement sensor combining cavity

tuning of a laser with a piezoelectric transducer’s subdivi-

sion technique is investigated. It has the merits of linear-ity, self calibration, and the simple structure which make

the sensor system less sensitive to the environmental dis-

turbance than a laser interferometer. A cat’s eye reflector is

used to improve the laser stability and a measuring range of 

100 mm is achieved. The piezoelectric transducer’s subdivi-

sion technique is implemented. Bidirectional sampling of ac-

tuated voltage on its rising and falling flanks amends the hys-

teresis and nonlinearity effect and the subdivision resolution

reaches 10 nm. The displacement sensor can measure the di-

mensions of precision work pieces such as MEMS, and the

method of cavity tuning of a laser and the piezoelectric trans-

ducer’s subdivision technique can be used in other measure-

ment fields.

ACKNOWLEDGMENTS

The authors are indebted to Matthias Dilger for polish-

ing work of this paper. We acknowledge the financial support

of the National Natural Science Foundation of China under

Grant Nos. 60827006 60723004.

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