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Smart Science Vol. 1, No. 1, pp. 1-12(2013)
Multi-DOF Incremental Optical Encoder with Laser Wavelength
Compensation
Cha’o-Kuang Chen1, Chien-Hung Liu2,* and Chung-Hsiang Cheng3
1Department
2Department
of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC
of Mechanical Engineering, National Chung Hsing University,Taichung,402, Taiwan, ROC
of Mechanical Engineering, National Taiwan University, Taipei, Taiwan, ROC
* Corresponding Author / E-mail: [email protected]
3Department
KEYWORDS : Multi-DOF, Laser optical encoder, Diffraction theory; Interference technique, Principle of auto-collimator, Wavelength monitoring
This study used a reflective diffraction grating as the medium to develop a multi-DOF incremental optical encoder for
motion stage. The optical encoder can measure three angular displacements, roll, yaw and pitch of the motion stage
simultaneously, as well as the horizontal straightness and linear displacement, summed to five DOF errors of motion stage
by only using the positive and negative first-order diffracted light. The grating diffraction theory, Doppler effect, and optical
interference technique were used. Two quadrant photodetectors were used to measure the changes in three-dimensional
space of diffraction direction of diffracted light, in order to construct a multi-DOF incremental optical encoder. Considering
the working stability of a laser diode and preventing the influence of the zeroth-order diffracted light returning to the laser
diode, an additional optical isolation system was designed and a wavelength variation monitoring module was created. The
compensation for the light source wavelength variation could be 0.001 nm. The multi-DOF verification results showed that
the roll error is ±0.7/60 arcsec, the standard deviation is 0.025 arcsec; the yaw error is ±0.7/30 arcsec, the standard deviation
is 0.05 arcsec; the pitch error is ±0.8/90 arcsec, the standard deviation is 0.18 arcsec, the horizontal straightness error is
±0.5/250 μm, the standard deviation is 0.05 μm and the linear displacement error is ±1/20000 μm, the standard deviation is
12 nm.
Manuscript received: August 09, 2013 / Accepted: September 18, 2013
1. Introduction
Renishaw [1], Heidenhain [2], Sony [3] and MicroE [4] are most
representative. The optical encoder is mounted inside the machine
tools, and the optical alignment technology can improve the
positioning accuracy of machine greatly, or proper feedback control
can improve the machining accuracy effectively. The disadvantage is
that the displacement information of only one linear axis can be
provided, thus resulting in various limitations in functions.
Many studies have been conducted on optical encoder design and
signal processing technique in early stages. In 1981, Heydemann [5]
successfully calculated the phase difference, amplitude error and zero
drift of two non-ideal sine wave interference signals by using the
least-squares fitting method and mathematical optimization. Practical
compensation was also made to increase the resolution and accuracy
of interferometer or interference system. In 1986, Akiyama et al. [6]
deposited metal on glass to make it with both functions of light
splitting and changing phase. It could replace λ/4 wave plate and
polarization plate in general interference system, and the overall
system was simplified. When a simple optical path design was
adopted, the orthogonal interference signal could be generated for
identifying the displacement and direction. In 1987, Taniguchi et al.
[7] of Japan Sony Corp. proposed a dual-beam incidence optical
encoder, its advantage was that it had theoretically infinite tolerance
in mounting distance, but its disadvantage was that it strongly limited
Both linear transmission and rotary transmission have six DOF
errors. In the ideal case, these transmission components are not
allowed to rotate in other five DOF directions when they move,
following a certain DOF. Thus, multi-DOF error may occur in stage
motion. In actual stage motion, if these six DOF errors are substituted
in the stage motion error model, the relative errors amount of laser
sintering process, cutter or measuring probe on the workpiece can be
obtained, and off-line or on-line error compensation can be carried
out. At present, in the development of intelligent precision mechanical
system as multi-axis machine tools, an important part is how to
microminiaturize and systematize these measuring modules and
integrate them into the machine tools. These sensors can detect multiDOF error in the machine tool, thus allowing the machine tools to
detect its kinematic errors instantly, and implement self-compensation
and machining state monitoring. However, the present measurement
systems, which can implement microminiaturization and
systematization and can be mounted inside machines for on-line realtime measurement, are mostly optical encoder for measuring linear
displacement and rotation angle displacement. Therefore, the optical
encoder is one of the most important instruments in machine tools or
precision motion stage. The products of large manufacturers, such as
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the deflection of grating. If unilateral light beam was disturbed by the
blot of grating or the process, the system could not make
compensation. Another Japanese company, Canon Inc. also
extensively studied optical type encoder. In 1990, Ishii et al. [8]
proposed laser power back off based on real-time laser intensity
variation monitoring and electronic circuit feedback, so as to correct
the variation of interference signal resulted from grating process error
or environmental fluctuation. The system stability could be improved
effectively. In the ensuing year, Ishizuka et al. [9] improved the
previous optical system, used focusing lens and aperture. The
unnecessary diffracted light could be isolated successfully. In the
same year, Nishimura et al. [10] used special three-dimensional
optical path design to avoid the ghost light of zeroth-order diffracted
light to the detector in optical dislocation mode, so as to increase the
signal to noise ratio. Following the previous improved optical system,
Ishizuka et al. [11] proposed a rotary laser optical encoder which
could improve the alignment tolerance. It suppressed the deflection
resulted from machine operation on the principle of optical lens. This
method could extend the application range of optical encoder greatly
by measuring the variation of optical path or signals. Besides Canon
Inc., Spies et al. [12] of Germany Heidenhain proposed an optical
encoder using non-polarization interference technique. The light
beam carrying displacement information was diffracted in the grating
multiple times, and superposed based on the grating diffracted light
characteristics, so as to generate interference fringe. The optical
system was formed simply. In the same year, Masreliez [13] of Japan
Mitutoyo successfully eliminated the diffraction efficiency variance
resulted from different polarization types of lightwave entering the
grating, improve the stability and accuracy of optical encoder greatly,
and improved the manufacturing tolerance of grating. The precision
positioning could reach a nano level. In 1995, Lin et al. [14] used λ/4
wave plate to change the polarization state, thus generating a set of
interference signals differing by 90°, and obtaining the displacement
and direction. The band pass filter and electronic interpolation
technology were used to purify the interference signal, the resolution
was 0.1 μm, and the stability was ±1μm/40min. In the same year,
Mollenhauer et al. [15] developed an interferometer allowing grating
deflection. Its principle was that the light beam entered the twodimensional grating to generate diffraction in horizontal and vertical
directions. There were four diffracted light beams, which entered the
two-dimensional grating to be measured. The interference was
generated after one more diffraction. In the optical principle, this
interference fringe would not change though the deflection of grating
was measured, so that the tolerance was improved, and the ghost
patterns of multiple diffractions were eliminated by minute angle
variation in assembly. Chiang of U.S. International Business
Machines (IBM) and Professor Chih-kung Lee of Taiwan University
[16] proposed an optical encoder for data recording disk drive servo
writing system in 1995, and used two sets of 1x telescopes for
wavefront reconstruction optics. The system could carry out
wavefront compensation for the deflection angle of disc grating and
the optical aberration resulted from radial grating. In 1996, Mitchell
et al. [17] of U.S. MicroE successfully used the optical path design of
natural interference to complete the optical encoder pickup head.
They indicated that the tolerance of pickup head for grating scale
could be improved greatly by using wavefront compensation. In 1999,
Henshaw [18] of Britain Renishaw added a medium with refraction
property to change the equal optical path of the interference system to
different the optical path. The error sensitivity between grating and
pickup head could be reduced effectively. In the same year, Sawada et
al. [19] successfully integrated the optical encoder pickup head into
the same GaAs substrate by using lithography, and completed a micro
encoder. Its size could be reduced to 0.5 mm3 (about 1/100 of general
optical encoder), and it had 0.01 μm resolution. It could be mounted
inside mini stages or micro motors for precision positioning, but its
alignment tolerance could not be improved, and it was difficult to be
mounted. In 2000, Kuroda [20] of Japan Sony Corp. proposed an
optical encoder pickup head system with high tolerance. The primary
diffracted light reached the fixed point when it returned to the grating
based on lens principle, and the secondary diffraction made the light
beams superposed to generate interference, so that the interference
light beams could be superposed forever. This system can resist the
vibration of machine tool effectively, and the optical path design can
avoid the diffracted light returning to the laser again to cause selfmixing problem, so that the reliability and accuracy of detection are
improved greatly. In 2001, Lee et al. [21] developed a new type of
diffractive laser optical encoder system. The optical system was
consisted of circular polarization interference, solution phase system
and 1x telescope. The measuring basis could be changed from laser
wavelength to grating pitch through non-contact optics. Therefore, it
had the advantage of anti-ambient interference, and the 1x telescope
structure could keep the diffracted light parallel to the incident light,
so that the alignment tolerance between optical head and grating was
increased by 6 to 20 times, as compared with the first-class optical
encoder in the world. In 2009, Liu et al. [22] proposed a new type of
multi-DOF optical encoder. It used linearly polarized He-Ne laser and
multiple four quadrant photodetector signals and polarization
interference signals to measure three angular errors and one
straightness error signal simultaneously, while measuring the
displacement. However, this optical system structure is large, it is not
suitable to be embedded in machine, and its cost is high relatively.
Based on the above studies, this study proposes the multi-DOF
optical encoder measurement system, which can detect the linear
displacement and three angular displacements, roll, yaw and pitch and
horizontal straightness of one shifting axle at the same time. The
system considered the working stability of a laser diode and
prevented the influence of the zeroth-order diffracted light returning
to laser diode. An additional optical isolation system was designed
and a wavelength variation measuring module was created for error
compensation of angular measurement.
2. Research and development of multi-DOF incremental
optical encoder with laser wavelength compensation
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The encoder developed in this study is shown in Fig. 1. It is
consisted of (1) optical isolator module with a polarized beam splitter
(PBS) and λ/4 wave plate (QWP), avoiding zeroth-order diffracted
light returning to laser diode to generate self-mixing phenomenon; (2)
wavelength variation monitoring module with a transmission grating
(TG), a focusing lens and one-dimensional position detector (1DPD)
to monitor the wavelength shift at any time, and then make
compensation; (3) four-DOF errors measuring system with two four
quadrant photodetectors (QPD1 and QPD2) by receiving positive and
negative first-order diffracted light; (4) phase measuring system with
three beam splitters (BS), a λ/4 wave plate (QWP), two polarizers (P)
and two photodetectors (PD 1 and PD2), and the overall system size
is 63×45×18mm3 (L×W×H).
the original optical path to cause interference.
2.2 Principle of wavelength variation monitoring module
The wavelength varies with time, temperature and power.
According to the diffraction angle variation analysis of grating
diffraction equation, the diffraction angle of positive and negative
first-order diffracted light varies with the wavelength, and this
variation will be introduced into the calculation error of four DOF
measurements. According to the grating diffraction equation:
sin  

d
(1)
When the diffraction angle has changed, Eqn. (1) can be changed to:
 sin  
2.1 Principle of optical isolator module
When the laser diode emits collimated laser to the reflective
grating vertically, there must be a zeroth-order diffracted light
generated along the original optical path, and this zeroth-order
diffracted light will return to the laser diode resonator, so as to
generate mixing phenomenon. Fig. 2 shows the designed optical
isolation module. The PBS splits the light beam into reflected s
polarized light and transmitted p polarized light. The reflected s
polarized light passes through λ/4 wave plate and then enters the
reflective grating vertically. It generates zeroth-order diffracted light
along the original optical path, and this return light beam will pass
through λ/4 WP again. Therefore, the zeroth-order diffracted light
polarization state will change from s polarization to p polarized light,
and the zeroth-order diffracted light cannot enter the laser diode along

1
  2 d
d
d
(2)
So the diffraction angle variation Δθ can be expressed as:
 
1 
1
(
  2 d )
cos d
d
(3)
Therefore, the variation of diffraction angle is influenced by the
wavelength variation Δλ and the grating pitch variation Δd, and the
wavelength variation influences the diffraction angle more
significantly. If the wavelength variation is 0.25 nm, and the laser
wavelength stability can be known from the laser wavelength
variation ratio Δλ/λ. If the relative variation rate Δλ/λ of the laser
diode used in this system is about 3.8×10-4, this wavelength variation
will result in 12.89 arc sec diffraction angle variation.
QPD2
QPD1
1DPD
BS
QWP
BS
Lens
LD
TG
PD1
PD2
P
QWP
PBS
RG
Fig. 1 Design drawing of multi- DOF laser optical encoder with laser wavelength compensation
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0 order
s
Fig. 2 Optical isolation module
Fig. 3 Wavelength variation monitoring module
Considering the influence of the grating scale thermal expansion Δd
on the diffraction angle, the quartz glass thermal expansion
coefficient is used for calculation. The grating pitch is 4μm, the
generated thermal elongation is 0.5 μm when the temperature change
per meter is 1 ℃, allocated to grating pitch averagely. The variation
Δd is about less than 2×10-6 μm, and the generated diffraction angle
variation is 0.017 arc sec. This influence is much slighter than
wavelength variation, so the center wavelength variation is the major
factor for the diffraction angle error.
The wavelength variation monitoring module is on the principles
of Spectroscope and Auto-collimator, and uses one-dimensional
transmission diffraction grating (TG), focusing lens and onedimensional position detector to form the wavelength detection
device. When the laser diode wavelength shifts, the diffraction angle
is changed due to the wavelength variation, and the wavelength
variation can be calculated by measuring the diffraction angle
variation. As shown in Fig. 3, the laser wavelength variation
monitoring module calculates the wavelength variation according to
the first-order diffracted light diffraction angle variation resulted from
wavelength shift. When the laser diode emits a collimated laser light,
and then the laser passes through the PBS, the transmitted p polarized
light passes through a transmission grating (TG). The generated firstorder diffracted light then enters the diffraction angle measuring
module. The diffraction angle variation can be calculated by Eqn. (4).
(4)
p x  fθ
where px is the light spot position variation, f is the focal length of
lens and Δθ is the diffraction angle variation. Therefore, the influence
of wavelength variation can be fed back by the wavelength variation
monitoring module to the four DOF error computing equation for
compensation and correction.
2.3 Four-DOF errors measuring system
Various coordinate systems must be determined before the
derivation by mathematical model of error, as shown in Fig. 4. First,
the reference coordinate system is defined as R  , and the light
source of this system is the collimated laser emitted from LD. The
reflector (M1) leads the light beam to the reflective grating (RG), so
the reflection point is defined as the light source coordinate system
S , and the coordinate point p go is 0 0 z  . The incidence
point of light source on the grating is defined as grating coordinate
system G, and the coordinate point p go is 0 0 0 ; the four
quadrant photodetector is defined as D m , m is the order number of
the diffracted light corresponding to its position. Therefore, the
R
transition matrix TG between grating coordinate system G and
reference coordinate system
R
can be obtained from the
homogeneous coordinate conversion, expressed as:
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R
 RR
TG   G
 0
R

pG 

1 
coordinate system
multiplying the rotational matrices of defined α, β and γ angles along

Dm
x, y and z axes. p G is the translation vector between grating
coordinate system G and reference coordinate system R. So the
TG between grating coordinate system G and
reference coordinate system R can be expressed as:
R

TG 
 c  c
c  s

 -s

 0
s s  c - c s
s s  s  c c
s c 
0
c s  c  s s
c s  s -s c
c c 
0
px 
py 

pz 

1
(6)
If
y

uS
z
T

uS are expressed as
G

uS x ,
G

uS
y
and
G

uS
z

u bm is shown in Eqn. (9), so the unit vector
can be obtained. Dm
light source coordinate system S and grating coordinate system

 
1-  G uS

y
G

u bm
z
2
x
m 

 d 

G

uS

1

2
y

1

T
Dm
Dm

1
z
(11)
T
x
and Dm
x Dm
y

u go
z

u go
z
z
y Dm
can be expressed as:
(12)
z
Laser diode emits a collimated laser beam into a one-dimensional
reflecting diffraction grating driven by the moving stage in the linear
displacement (Δx) will arise the signal with phase change generated
from the positive and negative first-order diffraction light due to the
Doppler shift relations. As shown in Fig. 5, the two reflective lights
will generate two sets of transmitted and reflected light based on the
beam splitter. The transmitted and reflected light overlap and interfere

The unit vector u go between grating coordinate point p go and
four quadrant photodetector coordinate system D m  can be

u go , and the

uG
ment)
Dm
R

uG can be
2.4 Phase measuring system (Linear displacement measure
(9)
T
obtained from the product of transition matrix
Dm
The Four-DOF errors can be calculated using three-dimensional
space of diffraction direction analysis and the inverse kinematic
analysis by mean of Eqn. (12).

u bm of m
order diffracted light can be expressed as:
G
u bm 
 
m G 
 G uS x 
uS
d



 G u bm x G u bm y
Dm
Dm 
u

Dm x  Dm u go x  Dm G
uG
Dm 
uG

Dm y  Dm u go y  Dm 
uG
When the system light source reaches the grating, the m order
G
diffracted light is generated, by using the unit vector u S between
G

u bm of m
Finally, the relation between the four quadrant photodetector
corresponding to the m order diffracted light and the grating actuation
(8)
G and the grating diffraction equation, the unit vector
G
, the
when the system light source irradiates the grating, expressed as:

T
u S  0 0 - 1 1


uG  Dm R R R R G G u bm


 Dm uG x Dm uG y

matrix representation can be carried out on this principle. The
G
R 
R
rotational matrix R R is the inverse matrix of R G . u S is the
unit vector of incident light in the reference coordinate system R 
R
R G and the unit vector
expressed as:
For convenient substitution in the future, x, y and z components of the
G
R
R R and
D m  . R R G is the rotation matrix between
grating coordinate system G and reference coordinate system R.
Dm
unit vector
Dm
coordinate systems
(7)

1

D
G
G
(10)
order diffracted light. m R R represents the rotation matrix between
reference coordinate system R and the four quadrant photodetector
expressed as:

 x cos  z sin    0 
 0 
0
 
x sin   z cos   p z 
 
1
 1 
T
Dm 
u go z 1

uG represents the unit vector between the m order diffracted
rotation matrices

The unit vector uS between light source coordinate system S
and grating coordinate system G can be obtained by multiplying
R
the rotational matrix G R R by the unit vector u S of incident light,


uS G R R R uS


 G uS x G uS
Dm
0 sin 
1
0
0 cos
0
0
Dm 
u go y
light derived from grating and the four quadrant photodetector
coordinate system D m  , it can be obtained from the product of
G
G
and the quadrant photodetector
Thus, it contains a rotational matrix


u go  Dm TR R u go
 cos
 0

  sin 

 0

 Dm u go x
R
transition matrix
D m  .
revolving round y-axis and a translation matrix, and the unit vector
Dm 
u go can be expressed as:
R
where R G is the rotation matrix between grating coordinate system
G and reference coordinate system R , it can be obtained by
R
R 
reference coordinate system
(5)
TR and unit vector
TR is expressed as the transition matrix between
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with each other, producing interference light beams which pass nonrotation (0 °) and rotated 90 ° of polarizer (P) respectively; therefore,
light intensity detectors (PD1 and PD2) can receive a set of
orthogonal interference signal. The two interference signals will
generate sinusoidal function of the changes in the strength (ie, sinθ
and cosθ) according to the movement of moving stage. Finally, the
actual linear displacement of the stage can be learned by determining
the changes of the two interference signals.
0 0 1 0
E PBS,s  
    
0 1 1 1
(14)
The s polarized light will pass the quadrant wave plate (QWP) in a
rotation of 45° and polarization state changes with the angle of
rotation, which can be expressed as:
E QWP(45 ) 
2.4.1 Linear displacement measurement for Jones Matrix
derivation
1  1  i   0 1   i 

  
 
2   i 1   1
21
(15)
The optical polarization of the system in the linear displacement
measurement was analyzed by Jones Matrix method in this section.
The initial electric field amplitude of the horizontal polarization of
laser light and vertical polarization of laser light( and ) are equivalent
for easily expressing and calculating; which can be expressed as:
The positive and negative first-order diffraction light is produced
while leading the beam into reflective grating. The zero-order
diffracted light (return light) will be separated by the optical isolator
module. The positive and negative first-order diffraction light
generated from Doppler effect of reflective grating diffraction light
can be expressed as:
 E x  1
E LD      
 E y  1
E 1 
(13)
The polarization state will change after passing the polarization beam
splitter (PBS), which can be expressed as:
E -1 
i
1   i  i
1  ie 
 i 
 1 e 
2 
2 e 
1
2
 i  i
1  ie  i 



 1 e
2  e i 
 
Fig. 4 Coordinate definition of four-DOF errors measuring system
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(16)
(17)
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Fig. 5 The phase measurement system
The negative first-order light will pass the quarter wave plate (QWP)
and (28), the phase can be calculated to determine the linear
displacement.
in a rotation of 90° again and bring phase retardation; therefore, the
change of negative first-order diffracted electric light field E -1 can be
expressed as:
E QWP(90 )
1 0  1  ie  i  1  ie i 






2  e i 
2   ie i 
0  i 
3. The verification result of the multi-degrees-of-freedom
increment optical encoder
(18)
Fig. 6 showed the photograph of the multi-degrees-of-freedom
increment optical encoder which adopted the internal lens of DVD
optical readhead. The mounting position should be in accordance with
corresponding location between the encoder and grating. When
thesystem light emitted from the laser diode (LD) and reflected by
reflection grating (RG), resulting the positive and negative first order
diffracted light. Through the interference optical element, the
diffracted light will be received by photodiode detector PDa and PDb,
and then apply the linear displacement measurement. The concept of
four degrees of freedom error measurement is that the incident
position of positive and negative first order diffracted light generated
by detector QPD1 and QPD2 is estimated and use Matlab software to
inversely solve the four degrees of freedom error quantity generated
by grating. This wavelength compensation module consists of a
transmission grating and an angle measurement system to carry out
the real-time monitor and compensation for wavelength. The
verification of each degree of freedom will be compared by Agilent
55292A laser interferometer.
The two laser beams will be leaded into beam splitters respectively,
which can be expressed as:
1 1 0 1  iei 
1  iei 
E R1  






2 0 1
2  ei  2 2  ei 
(19)
E R2 
1 1 0 1  ie  i 
1   ie  i 






2 0 1
2   ie i  2 2   ie i 
(20)
The two reflective lights will convert into transmitted (T) and
reflective (R) light, thus E R1 and E R 2 of the electric field can be
expressed as:
E R1 ,T  E R1 ,R 
i
i
1 1 0
1  ie 
1  ie 

 i 
 i  


2 0 1  2 2  e  4 2  e 
(21)
E R 2 ,T  E R 2 ,R
1 1 0
1
 

2 0 1 2 2
 ie  i 
1  ie  i 



i 
i 
  ie  4 2   ie 
(22)
Also, the transmitted light and reflected light will overlap and
interfere together, generating interference light beams. The electric
fields can be expressed as: Eqn. (23) and Eqn. (24). The two
interference light beams will pass through polarizers in a rotation of
(0°) and (90°) respectively and will be converted
3.1 The wavelength variation monitoring module
The wavelength compensation module was combined by a
single angle measurement system which including a focusing lens)
and one-dimensional photodiode detector. From the wavelength
compensation module theory in the previous section, we can learn
that variation quantity is relevant to focal length instead of the
distance between light and detector when the source of incident light
is changed. This is the autocollimator principle, as described in Eqn.
(4). As also shown in Fig. 3, the wavelength compensation module
into E P1 0  and E P2 90  , which can be expressed as: Eqn. (25) and
Eqn. (26). E P1 0  and E P2 90  will be received by the photodetectors
(PD1) and (PD2). The light intensity will be proportional to the
square of the amplitude of the electric field vector. As a result, the
light intensity can be converted from two electric fields by the
following equations: Eqn. (27) and Eqn. (28). From two Eqns. (27)
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meet the wavelength shift detected by HR4000CG-UV-NIR
broadband spectrometer.
can detect the angle changes of positive first-order diffraction light by
transmission grating (TG) to convert the laser wavelength shift. Fig. 7
shows the shifting condition of wavelength for the system within
eight hours. At 200 min and 450 min, the wavelength shift results
E L1  E R 1 , R  E R 2 , T 
E L1  E R 1 , T  E R 2 , R 
1  iei 
1  ie  i 
1  iei  ie  i 





 i  
4 2  e  4 2  ie  i  4 2  ei  ie  i 
i
 i
i
(23)
 i
1  ie 
1   ie 
1   ie  ie 

 i  



 i 
4 2  e  4 2   ie  4 2  ei  ie  i 
i
 i
i
1 0
1  ie  ie 
1  ie  ie

E P1 0   

  4 2  i
 i 
0
0
0


 e  ie
 4 2
i
 i
(24)



(25)
 i
0
0 0

1  ie  ie 
1 
E P2 90   
  4 2  ei  ie  i   4 2 ei  ie  i 
0
1






I PD1 
I PD 2 
1
4 2
ie
i
 ie i

0
1   ie

4 2
i
 ie
0
 i
(26)
 1
  1  cos(2 )
 16
(27)
0
 1
1
1 
 1  sin(2 )
0 e i  ie i 
i 
 i
4 2
4 2 e  ie  16


(28)
(a) The inner setup of optical elements
(b) The overall system diagram
Fig. 6 The photograph of multi-degrees-of-freedom increment optical encoder with laser wavelength compensation
652.708 nm
652.306 nm
Fig. 7 The wavelength shift detection (Time: 8 hrs)
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Fig. 8 The calibration of four quadrant photodetector (QPD1)
Fig. 9 The calibration of four quadrant photodetector (QPD2)
3.2. The calibration of two quadrant photodetectors
55292A laser interferometer: The residual error is ±0.7 arcsec/30
arcsec, and standard deviation is less than 0.05 arcsec, as shown in
Fig. 11.
The calibration results of two quadrant photodetectors (QPD1)
and (QPD2) are shown in Fig. 8 and Fig. 9.The calibration range of
the quadrant photodetector is ± 40 μm, the residual error is ± 0.25 μm,
and standard deviation is 0.012 μm
3.5 The verification of pitch (θz) angular error of moving
stage:
The calibration range of pitch angular error is ±90 arcsec along zaxis. The measurement results were compared with the Agilent
55292A laser interferometer: the residual error is ±0.8/90 arc sec, and
standard deviation is less than 0.18 arc sec, as shown in Fig. 12.
3.3 The verification of roll angular error (θx)
A swing-axis stage was used to generate the rotational angle (θx)
with ±60 arcsec along x-axis. The measurement results were
compared with the Agilent 55292A laser interferometer: the residual
error is ±0.7 /60 arc sec, and standard deviation is less than 0.025 arc
sec, as shown in Fig. 10.
3.6 The verification of straightness error (Δz)
A linear stage was used to generate straightness error (Δz) within
±250 μm. The measurement results were compared with the Agilent
55292A laser interferometer: the residual error is ±0.5/250 μm, and
standard deviation is less than 0.05 μm, as shown in Fig. 13.
3.4 The verification of yaw angular error (θy)
The calibration range of rotational angle (θy) is ±30 arcsec along
y-axis. The measurement results were also compared with the Agilent
(a) Residual error
(b) Standard deviation
Fig. 10 The verification of roll angular error
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(a) Residual error
(b) Standard deviation
Fig. 11 The verification of yaw angular error
(a) Residual error
(b) Standard deviation
Fig. 12 The verification of pitching angular error
3.7 The linear displacement measurement (Δx) of moving
stage:
This study used the automatic linear stage to move displacement
amount (Δx) with ±20000 μm. The diffracted light will be received by
light intensity detector PD1 and PD2 in the readhead, and then
calculate the linear displacement. The stability detected in the system
showed the standard deviation is 12 nm in Fig. 14. The measurement
results were compared with the Agilent 55292A laser interferometer
and the residual error is ±1/ 20000 μm, as shown in Fig. 15.
intelligent machine tool. The multi-degrees-of-freedom laser encoder
system developed by this study has already completed miniaturization
development. The multi-axis machine tools installed the proposed
systems can carry out multi-axis (x, y, and z-axis) errors detection for
compensation. Taking three-axis (x, y, and z-axis) processing machine
tool as an example, it can detect the 15 errors generated by three
linear axes if this system is installed with the proposed system. The
function of this system is higher than those traditional optical
encoders with the same mounting method on the linear axis in
machine tools.
5. Conclusions
ACKNOWLEDGEMENT
This paper provided a new development direction for the
application of optical encoder. It could be applied to machine tools in
order to realize the self-measurement, compensation and diagnosis of
geometrical error of space, and shorten the time for correction by
laser interferometer, so as to develop a self-error compensating
The authors gratefully acknowledge the financial support provided to
this study by the National Science Council of Taiwan under Grant Nu
mber NSC 100-2221-E-005-091-MY3.
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(a) Residual error
(b)Standard deviation
Fig. 13 The verification for straightness measurement
Fig. 14 System stability test
(b) Residual error
(a) Linearity and repeatability
Fig. 15 The verification of linear displacement
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