2009_sci_correction of field distortion of laser marking systems using surface compensation function
TRANSCRIPT
Correction of field distortion of laser marking systems using surface
compensation function
Ming-Fei Chen, Yu-Pin Chen �, Wen-Tse Hsiao
Department of Mechatronics Engineering, National Changhua University of Education, No.1 Jin-De Road, Changhua City 50058, Taiwan
a r t i c l e i n f o
Article history:
Received 18 December 2007
Received in revised form
10 May 2008
Accepted 30 July 2008Available online 27 September 2008
Keywords:
Field distortion
Laser marking system
Compensation
Galvanometric scanning system
a b s t r a c t
Laser scanning systems are extensively applied in laser marking, laser printers, laser projection, and
laser coding. They represent a very mature technique in the marking industry, and the galvanometric
scanning systems are usually adopted in the laser marking system. Although the galvanometric
scanning system usually increases the marking speed, it is usually accompanied by field distortion. The
paper gives a correction method for the field distortion of a laser marking system, which, using the
surface curve fitting function, corrected the control system of the scanning system. The field distortion
errors of the laser marking system are corrected by the surface compensation function, which, using the
surface curve fitting method, obtained the corrected position of laser spots. The results in this paper
indicate that field-distortion errors of laser marking systems are effectively corrected by the surface
compensation function. Moreover, the compensation method would be widely adopted to increase the
accuracy of most two-dimensional machine systems, such as XY table, etc.
Crown Copyright & 2008 Published by Elsevier Ltd. All rights reserved.
1. Introduction
Laser marking is a rapid, non-contact means of producing
permanent high-resolution images on the surface of most
engineering materials. The most popular sources are pulsed CO2
laser, Nd:YAG laser and Excimer lasers with average power levels
of several tens of watts. The laser beam is scanned over the
materials using computer-controlled scanning systems, or pro-
jected through a mask or stencil to generate the images. Laser
scanning methods are widely used in the laser marking processing
because the scanning method provides flexibility in character or
image generation. A few types of laser scanning systems have
been applied in the laser marking system, including the oscillating
mirror, the mutating mirrors, the fiber scan, the rotating polygon
and the acousto–optic deflector (AOD), as shown in Fig. 1. Nearly
all laser marking processings are used by oscillating mirrors,
which is usually called the galvanometric scanning system. The
laser galvanometric scanning method provides an effective
solution in many image display applications, including medical
imaging, laser display, laser marking and materials processing.
Galvanometric scanners combine a mirror with a servo-
actuator limited-rotation motor, which is the main component
in the galvanometric scanning system. They control the laser
beam onto the defined locations via turning scanning mirrors.
Therefore, the mirrors can perform high-speed deflection about an
axis of rotation of the galvanometric scanning motor. The single
galvos only can deflect a laser beam along a line in an image plane.
The galvanometric scanning system and the position of the laser
beam must be controlled, both horizontally (X-axis) and vertically
(Y-axis), to create an image with two axes. XY scanners deflect a
parallel beam in the X–Y direction. The laser beam enters the
system through an input aperture. Once inside, the laser beam
hits the first mirror on the X scanner and then the second mirror
on the Y scanner deflects it. Accordingly, the laser beam would be
projected on the defined image field. Scanning systems are of two
general types [1]. The first is the pre-objective scanning system, in
which the XY scanners are located in front of an F-theta focusing
lens, as shown in Fig. 2(a). The other is the post-objective
scanning system, in which the XY scanners are located behind the
focusing lens, as shown in Fig. 2(b).
Scanning accuracy is an important parameter in the scanning
system, but it is influenced by the errors that cause image
distortion in the scanning system. Xie et al. [2] reported a
correction of image distortion for a laser galvanometric scanning
system and utilized the correction algorithm to reduce the
distortion error of scanning images. The correction is made
by predicting the distortion error of the scanning system using
an F-theta lens to correct the focused image. Although it could
effectively increase the precision of the focusing image, it could
not correct all errors. An error algorithm in machine systems
fails to predict various errors because the errors are irregular.
Hence, the compensation methods could correct all the errors of
ARTICLE IN PRESS
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/optlaseng
Optics and Lasers in Engineering
0143-8166/$ - see front matter Crown Copyright & 2008 Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.optlaseng.2008.07.017
� Corresponding author. Tel.: +88647232105x7283; fax: +886 47211149.
E-mail addresses: [email protected] (M.-F. Chen),
[email protected] (Y.-P. Chen), [email protected] (W.-T. Hsiao).
Optics and Lasers in Engineering 47 (2009) 84–89
the machine system using an error function from the original
errors of the workpiece that were manufactured by the machine
system. Chen et al. [3] have provided a compensation method of
field distortion in the CO2 laser drilling system and utilized the
compensation function to reduce the distortion errors and
position errors of laser drilling systems. The compensation
function is obtained by the original errors of the scanning system
using the Lagrange interpolation method. Although Lagrange
interpolation can effectively reduce the errors of the scanning
system, the non-continuous result of the scanning image would
be accompanied in the marking pictures.
The distortion of the laser marking system is a two-dimen-
sional error in the focusing plane, which could be fitted by a
surface and a two-degree polynomial. Many mathematical
algorithms can fit a surface, including the Cubic Splines, the
Bezier surface, the B-Spline surface, and others [4]. The paper
gives a common algorithm that is a computed surface function by
nine points, and corrected the distortion error of the scanning
image using the surface error function.
2. Error analysis of the galvanometric scanning system
The galvanometric scanning system is widely used in laser
marking systems because the scanning system provides high
marking speed and fixable processing. However, the field-
distortion error is usually accompanied in the scanning system.
Hence, although the scanning system was supposed to draw a
square, the actual drawing image would not be a square, it may
appear as a distortion image.
The distortion error of the laser marking system is different
from the common distortion aberration. The common distortion
aberration, which causes the projection image to be unlike the
objects, is generated by the aberration of optical elements.
Actually, the image of the laser marking system is generated by
the scanning points of the control system. Hence, the field
distortion of the laser marking system is generated by the laser
beam drift error and optical path error. These various sources of
errors that influence the scanning system will be discussed.
2.1. Laser beam drift errors
These various errors usually occur as laser beam drift errors in
the laser scanning system, including the mechanical misalign-
ment, the thermal effect, and the assembly errors [2]. Several
mechanical misalignments are produced by fixing the scanning
head, which results in the laser beam drift. Thermal effect will
also be the source of the beam drift errors in the laser marking
system. Moreover, the position transducer of the scanning system
is sensitive to the change in temperature.
ARTICLE IN PRESS
Rotation
Mirror
Input laser beam
Reflecting beam
Mirror
ω
φ
α
Rotating polygon
ω
Input laser beam
Reflecting beam
Input laser beam
Reflecting beam
AOD
Input laser beam
Bra
gg a
ngle
First order
Zero order
Input laser beam
Fiber Switch
Fig. 1. Types of laser scanning systems [1].
Scan Mirror
Laser beam
focal lens
Laser beam
focal lens
Scan Mirror
f
y
θ
Fig. 2. Types of scanning systems [3]. (a) Pre-objective scanning and (b) post-objective scanning.
M.-F. Chen et al. / Optics and Lasers in Engineering 47 (2009) 84–89 85
2.2. Optical path errors
The distortion error does not result in a blurred projection
image. In the absence of any other aberration, distortion is
manifest in a misshaping of the image as a whole, even though
each point is sharply focused. Fig. 3 shows a scanning systemwith
F-theta lenses, and it indicates the origin of distortion. The laser
beam incident at an angle on the F-theta lens, passes through it,
such that a reasonable difference arises between the paraxial
angle of refraction and the real ray angle of refraction. Restated,
distortion arises because different areas of the lens have different
focal lengths and different magnifications. In this case, this caused
the real image to be pulled inward from the paraxial image thus
causing negative or barrel distortion. The amount of distortion is
expressed either as a lateral displacement in units of length units
or as a percentage of the paraxial image height in an image plane.
It is defined as [5]
Distortion ¼yÿ ypyp
� 100% (1)
where y is the height of the image plane and yp is the paraxial
height.
The extent of distortion may be positive or negative. A few
distortions in the galvanometric scanning system, including
pillow-shaped, barrel-shaped and pillow-barrel-shaped image,
are because of optical path error, as illustrated in Fig. 4. Although
the scanning system was supposed to draw a square, the actual
image was not a square. A scanning system suffering positive
distortion deforms a square array, as shown in Fig. 4(b). In such a
situation, each image point is displaced radially outward from the
center, and the most distant points move farthest. For negative
distortion, each point on the image moves radially inward toward
the center, as shown in Fig. 4(c). Positive distortion typically
results in a pillow-shaped image in the image plane; negative
distortion results in a barrel shape. Although the distortion did not
affect the results of the projected image, it was associated with a
defect aberration and an unacceptable image in the visual system.
Distortion of the order of 2–3% is typically an acceptable visual
system.
Most laser galvanometric scanning systems for laser drilling
machines use a multi-element F-theta lens to amend the
projected image plane and the scanning velocity. In such a case,
the distortion represents a great error in the scanning system,
because the multi-element lens system greatly increases distor-
tion. Therefore, correcting the distortion of the scanning system in
a CO2 laser drilling machine is very important. In a CO2 laser
drilling machine, the distortion error is a pillow-barrel-shaped
distortion of the square image field, which is caused by the
path of the beam in the scan head and by the objective. The
control can compensate for this field distortion, and the method of
compensation is presented herein.
3. Correction of field distortion
Field-distortion errors in a scanning system normally consist of
systematic errors and random errors in the scanning system.
A systematic error is defined as the manufacturing error and the
assembling error. It is repeated in the mechanical systems.
Systematic errors of a mechanical system, which were usually
obtained by the theoretical analysis and measurement, can
be reduced using the compensation techniques. Although the
random error was not a repeated error, it could also be
compensated for by predicting errors in the system. Accordingly,
the error compensation technique is a widely accepted method for
correcting machine errors, and was effective in precision machine
design and manufacture. This work gives a normal surface
compensation function, which is consisted of nine points, to
amend the field-distortion error of laser marking systems.
The field distortion of a laser marking system is a two-
dimensional error function in the focusing plane. Methods such as
interpolation and curve fitting can be used in obtaining the
two-dimensional error functions. The investigation used the
surface curve fitting method to obtain the compensation function
and correct the field distortion of the laser marking system.
The two-dimensional polynomial function z is a polynomial
function of two variables, x and y—say, of degree 2 each of in x
and y. That is
Z ¼ f ðx; yÞ ¼ a0 þ a1xþ a2yþ a3xyþ a4x2
þ a5y2 þ a6x
2yþ a7xy2 þ a8x
2y2 (2)
The function describes the surface; (x, y and z) is a point on it.
The coefficients of the aforementioned function were obtained by
solving this equation, which could be used to estimate the value
of z (f(x, y)) at various values of x and y. The nine unknown
variables of equation can be solved using a set of equations as
ARTICLE IN PRESS
Fig. 4. Illustration of different distortions in an optical system [5]. (a) No distortion, (b) pillow shape, (c) ballew shape and (d) pillow-barrel shape.
F-theta lens
real chief rayparaxial chie fray
Image plane
location
resulting distortion
Fig. 3. Illustration of field distortion in the laser scanning system with an F-theta
lens [5].
M.-F. Chen et al. / Optics and Lasers in Engineering 47 (2009) 84–8986
shown in Eq. (3):
a0 a1 a2y1 a3y1 a4 a5y21 a6y1 a7y
21 a8y
21
a0 a1 a2y2 a3y2 a4 a5y22 a6y2 a7y
22 a8y
22
a0 a1 a2y3 a3y3 a4 a5y23 a6y3 a7y23 a8y
23
2
6
6
4
3
7
7
5
�
1 1 1
x1 x2 x3
1 1 1
x1 x2 x3
x21 x22 x23
1 1 1
x21 x22 x23
x1 x2 x3
x21 x22 x23
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
¼
z11 z21 z31
z12 z22 z32
z13 z23 z33
2
6
6
4
3
7
7
5
(3)
To solve the equation, we will obtain these coefficients of
variables. These coefficients of a0 to a8 are shown as the following
Eqs. (4)–(12):
a0 ¼ z11 ÿ ða1x1 þ a2y1 þ a3x1y1 þ a4x21
þ a5y21 þ a6x
21y1 þ a7x1y
21 þ a8x
21y
21Þ (4)
a1 ¼z22 ÿ z21x2 ÿ x1
ÿ y2a3 ÿ ðx2 þ x1Þa4
ÿ ðx2 þ x1Þy2a6 ÿ y22a7 ÿ ðx2 þ x1Þy22a8 (5)
a2 ¼z21 ÿ z11y2 ÿ y1
ÿ x1a3 ÿ ðy2 þ y1Þa5
ÿ x21a6 ÿ ðy2 þ y1Þx1a7 ÿ ðy2 þ y1Þx21a8 (6)
a3 ¼z22 ÿ z21 ÿ z12 þ z11ðx2 ÿ x1Þðy2 ÿ y1Þ
ÿ ðx2 þ x1Þa6
ÿ ðy2 þ y1Þa7 ÿ ðx2 þ x1Þðy2 þ y1Þa8 (7)
a4 ¼ P1 ÿ y1a6 ÿ y21a8 (8)
a5 ¼ O1 ÿ x1a7 ÿ x21a8 (9)
a6 ¼P2 ÿ P1
y2 ÿ y1ÿ ðy2 þ y1Þa8 (10)
a7 ¼O2 ÿ O1
x2 ÿ x1ÿ ðx2 þ x1Þa8 (11)
a8 ¼ðO3 ÿ O1Þðx2 ÿ x1Þ ÿ ðO2 ÿ O1Þðx3 ÿ x1Þ
ðx3 ÿ x1Þðx2 ÿ x1Þðx3 ÿ x2Þ(12)
where
O1 ¼ðz31 ÿ z11Þðy2 ÿ y1Þ ÿ ðz21 ÿ z11Þðy3 ÿ y1Þ
ðy3 ÿ y1Þðy2 ÿ y1Þðy3 ÿ y2Þ(13)
O2 ¼ðz32 ÿ z12Þðy2 ÿ y1Þ ÿ ðz22 ÿ z12Þðy3 ÿ y1Þ
ðy3 ÿ y1Þðy2 ÿ y1Þðy3 ÿ y2Þ(14)
O3 ¼ðz33 ÿ z13Þðy2 ÿ y1Þ ÿ ðz23 ÿ z13Þðy3 ÿ y1Þ
ðy3 ÿ y1Þðy2 ÿ y1Þðy3 ÿ y2Þ(15)
P1 ¼ðz13 ÿ z11Þðx2 ÿ x1Þ ÿ ðz12 ÿ z11Þðx3 ÿ x1Þ
ðx3 ÿ x1Þðx2 ÿ x1Þðx3 ÿ x2Þ(16)
P2 ¼ðz23 ÿ z21Þðx2 ÿ x1Þ ÿ ðz22 ÿ z21Þðx3 ÿ x1Þ
ðx3 ÿ x1Þðx2 ÿ x1Þðx3 ÿ x2Þ(17)
The relationship between the spot positions and the scanning
angle of the galvos is usually determined by adjusting the
scanning angle to focus at the define position on a specimen.
The correct scanning angle can be obtained, and the galvos can be
used to project the laser beam onto the defined location.
Therefore, the correct scanning angle of galvos can amend the
field distortion. According to Eqs. (2)–(17), the correct scanning
angles of the laser marking system are built, and the laser
marking system will amend the distorted error by this actual
scanning image. Fig. 5 simulated the several field distortions
of scanning image using the surface function. The simulation
results indicated that the distortion image was successfully
described by the surface function, as shown in Eq. (2).
4. Experiment and results
The correct scanning angles of the laser marking system
mentioned above were based on CO2 laser marking machines,
which were developed in the authors’ laboratory at the National
Changhua University of Education. Although the scanning system
causes field distortion, it is always adopted in laser marking
machines. Therefore, to develop, compensation function of field
distortion is the main task in the field of laser scanning systems.
The field distortion of a scanning image is typically asymmetric
about the centerline, and predicting the distortion error of the
scanning image is very difficult. This investigation provides a
technique for correcting the field distortion in the scanning image
ARTICLE IN PRESS
25
20
15
10
5
0
-5
-10
-15
-20
-25
Y-d
irection (
mm
)
25
20
15
10
5
0
-5
-10
-15
-20
-25
Y-d
irection (
mm
)
25
20
15
10
5
0
-5
-10
-15
-20
-25
Y-d
irection (
mm
)
-25 -20 -15 -10 5 0 5 10 15 20 25
X-direction (mm)
-25 -20 -15 -10 5 0 5 10 15 20 25
X-direction (mm)
-25 -20 -15 -10 5 0 5 10 15 20 25
X-direction (mm)
Fig. 5. Simulate the field-distortion error of the scanning system using MATLAB program. (a) pillow-shape distortion, (b) barrel-shape distortion and (c) pillow-barrel shape
distortion.
M.-F. Chen et al. / Optics and Lasers in Engineering 47 (2009) 84–89 87
by the surface compensation function of the correction scanning
angle of a laser marking system, as discussed below.
The position of the visible laser spots could be measured using
a photo-sensor, but not all commodity sensors can measure the
CO2 laser. Therefore, the position of the CO2 laser spots is typically
measured from the rectangular array holes of a workpiece, which
are drilled by the laser scanning system. The acrylic plate and
paper were selected as the compensation specimen, because of
their better adsorption in the CO2 laser. Although the positioning
accuracy of the galvanometric scanning system is usually
measured by a CCD image processing system, the CCD image
processing does not construct in all of laser marking systems.
Hence, the positions of laser spots of the laser marking system
cannot be measured using the CCD image processing system. In
order to measure the position of laser spots in the laser marking
system, the investigation uses the graph paper to measure the
position of laser spots. Although the measuring method, which
uses graph paper to obtain the position of laser spots, would
decrease the positioning accuracy of the laser scanning system,
the field distortion of the laser marking system would be
effectively reduced. Therefore, the graph paper would be adopted
in correcting the field distortion of laser marking systems.
The system parameters had a scanning area of 40� 40mm2. The
laser was a CO2 laser, with awavelength of 10,600nm and a power of
30W. The scanning system was a pre-objective system. Grids of
3�3 were established over the scanning field to eliminate field
distortion and the position error of the galvanometric scanning
system. The 9 points over the scanning field produced a surface
polynomial, and the polynomial was integrated into the control
program to adjust the scanning angle of galvos to project the laser
spots on the defined position. Fig. 6 schematically depicts the
program for correcting field distortion using a personal computer.
The procedure for compensating the field distortion is as follows:
1. The laser marking system project laser spots onto the defined
position on the graph paper, which adjusts the scanning angles
of the scanning system. The corrected scanning angle of the
laser marking system, which could project spots onto the
defined position, could be obtained.
2. Nine points of the scanning area produced the compensation
function for the laser marking system using the surface curve
fitting method, as shown in Eqs. (3)–(17).
3. The compensation function of the laser marking system is
integrated into the control system. The correct scanning angles
of the galvanometric scanning system correct the scanning
angle and eliminate the field distortion in the image plane.
4. A picture on the graph paper was marked using the CO2 laser
marking system, using the compensation technique to reduce
distortion errors.
The correct scanning angles of laser marking systems were found
and the surface compensation polynomial was obtained using the
surface curve fitting method to reduce field distortion of the scanning
image plane. Therefore, the laser galvanometric scanning system
could be integrated with the compensation program in a control
system that is based on a personal computer. The investigation
reduced the field distortion of laser marking systems using the
compensation method that integrated the correct scanning angle of
the scanner in the control system. According to this technique, the grid
pattern on the graph paper was marked using a CO2 laser marking
system with compensating field distortion, as shown in Fig. 7(b).
A comparison with Fig. 7(a) indicates that the compensation method
for field distortion improves the processing quality in a laser marking
machine, and the field distortion can be reduced effectively by the
compensative methods in the laser galvanometric scanning system.
In order to verify whether the compensation method can
effectively reduce the field distortion that resulted by a laser
galvanometric scanning system, the results of the laser marking are
made on the CO2 laser marking machine, as shown in Fig. 8. Fig. 8
ARTICLE IN PRESS
Start
To control scanning angle of
scanner focus laser spots onto
the definition position.
Nine points of corrected
scanning angles are established a
surface polynomial using the
surface curve fitting method.
Creating a correcting program
based on PC-Base Controller.
A picture is marked using the
laser marking system with
compensaton.
END
Fig. 6. Flowchart of the distortion compensation using the surface curve fitting
method.
Fig. 7. The square array of holes on the graph paper was marked by the laser
marking system. (a)Without compensation and (b) with compensation.
Fig. 8. Laser marking with field-distortion errors. (a) Pillow-shape distortion and
(b) Barrel-shape distortion.
M.-F. Chen et al. / Optics and Lasers in Engineering 47 (2009) 84–8988
shows a photograph of the various distortion errors that marked on
the graph paper using the galvanometric scanning drilling system,
including pillow-shaped and barrel-shaped distortion results. The
results verify that the technique can generate the various distortion
errors on the scanning image. The compensation procedures usually
use the opposite scanning angle to reduce the field distortion. Hence,
the position of focusing spots on the projection image could be
controlled by the compensation method in the galvanometric
scanning system. Fig. 9 shows the mascot of ChangHua University
of Education marked on a graph paper that used the compensation
method. The results of this experiment verified that the compensa-
tionmethod, described in this study, could effectively reduce the field
distortions in the laser marking images. The method of error
compensation is widely employed in scanning systems, such as the
laser drilling machines, the laser marking systems, the laser rapid
manufacturing systems and others.
5. Conclusion
This investigation proposes a method for correcting the field-
distortion error in a CO2 laser marking system. Field distortion
always occurs in a laser scanning system, and causes unacceptable
errors in galvanometric scanning systems. Reducing the field
distortion in a scanned image is very important to a laser marking
machine. Therefore, this investigation discusses the compensation
method to correct the field-distortion error in galvanometric
scanning systems.
Field distortion is an irregular and a two-dimensional error in
the scanning system and usually corrected using the compensa-
tion methods. The study develops a compensation method for
correcting field distortions of a scanning image. The compensation
technique is used in the surface curve fitting method to obtain
the function of correction control commands of the galvos, which
can compensate the distortion error in the galvanometric
scanning systems. Experimental results verified that the techni-
que could effectively reduce the field-distortion errors in the
laser drilling machine. Hence, the technique could effectively
decrease the field-distortion error of the galvanometric scanning
system, and can be widely adopted to increase the accuracy
of most of the two-dimensional machine systems, such as XY
table, etc.
Acknowledgment
The authors would like to thank the National Science Council
of the Republic of China, Taiwan, for financially supporting this
research under Contract no. NSC 96-2221-E-018-014.
References
[1] Wehr A, Lohr U. An introduction and overview ISPRS. J Phorogramm RemoteSensing 1999;54:68–82.
[2] Xie J, Huang S, Duan Z, Shi Y, Wen S. Correction of the image distortion for lasergalvanometric scanning system. Opt Laser Technol 2005;37:305–11.
[3] Chen MF, Chen YP. Compensating technique of field-distorting error for the CO2
laser galvanometric scanning drilling machines. Int J Mach Tools Manuf2007;47:1114–24.
[4] Gerald GF, Wheatley PO. Applied numerical analysis. New York: Addison-Wesley Publishing Company; 1996.
[5] Fischer RE, Tadic-Galeb B. Optical system design. New York: McGraw-Hill;2000.
ARTICLE IN PRESS
Fig. 9. Photograph of the mascot of ChangHua University of Education is marked
using the CO2 laser marking machine. (a) Original picture and (b) compensation
result.
M.-F. Chen et al. / Optics and Lasers in Engineering 47 (2009) 84–89 89