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Bo Tao
Department of Precision Machinery and
Precision Instrumentation,
University of Science and Technology of China,
Hefei, Anhui 230026, China;
Department of Integrated Systems Engineering,
The Ohio State University,
210 Baker Systems Engineering,
1971 Neil Avenue,
Columbus, OH 43210
Peng He
Department of Integrated Systems Engineering,
The Ohio State University,
210 Baker Systems Engineering,
1971 Neil Avenue,
Columbus, OH 43210
Lianguan Shen
Department of Precision Machinery and
Precision Instrumentation,
University of Science and Technology of China,
Hefei, Anhui 230026, China
Allen Yi
1
Department of Integrated Systems Engineering,
The Ohio State University,
210 Baker Systems Engineering,
1971 Neil Avenue,
Columbus, OH 43210
e-mail: yi.71@osu.edu
Annealing of Compression
Molded Aspherical Glass Lenses
Residual stresses and refractive index of molded glass lenses are important quality indi-
cators of their optical performance. In this research, the control of residual stresses and
refractive index variation of molded glass lenses were experimentally investigated by
postmolding annealing. Residual stresses were quantitatively measured using a circular
polariscope. Refractive index was reconstructed and calculated by an optical setup based
on Mach–Zehnder interferometer. In addition, geometry of the aspherical surface of lens
was also evaluated before and after annealing. The comparison between the measured
results before and after annealing showed that residual stresses and refractive index vari-
ation were well controlled and the shape of the aspherical surface was largely retained.
This comprehensive experimental study demonstrated a suggestion to improve quality of
the compression molded glass lens by postmolding annealing for high-precision optical
applications. [DOI: 10.1115/1.4025395]
Keywords: annealing, glass molding, residual stresses, aspherical glass lens, refractive
index variation, curve deviation
1 Introduction
Compression molding is a high-volume and precision fabrica-
tion method for glass optic elements [1–4], such as glass lenses
for digital projectors, light-emitting diode (LED) lightening, and
LED collimators. However, a small amount of residual stresses
[5] and refractive index drop [6,7] were often observed in molded
glass lenses. Residual stresses inside a glass lens can cause bire-
fringence and refractive index variation. Refractive index varia-
tion caused by both residual stresses and refractive index drop
will induce distortion to the wave front passing through the glass
lens. All these factors contribute to image quality degradation.
Therefore, control of residual stresses and refractive index varia-
tion is an integral part of precision compression molding
technique.
In compression molding process, cooling rate is carefully con-
trolled to maintain the residual stresses and refractive index varia-
tion below a required value. Slower cooling rates lead to lower
residual stresses [5] and less refractive index variation [8]. How-
ever, slower cooling rates used in the compression molding pro-
cess can also result in longer production cycle, thus reducing
production efficiency. For a proper production rate, the cooling
rate cannot be arbitrarily made too slow, especially in industrial
scale production. Therefore, a relatively high cooling rate in com-
pression molding process is often used for fabrication of molded
glass lenses and the molded glass lenses may not meet the require-
ments of precision glass lenses.
In order to ensure the quality of the compression-molded glass
lenses, a postmolding annealing process may be considered.
Annealing is a slow cooling process to relieve internal stresses
inside molded glass lens after molding. It has long been studied as
a process for improving the quality of optical glass [9–12]. The
structural relaxation model needs to be implemented to accurately
analyze the annealing process [13]. After annealing, quality of the
glass lens can be improved in two different ways: to achieve lower
residual stresses and better refractive index homogeneity. A major
advantage of annealing after compression molding is that multiple
molded glass lenses can be annealed together to save time.
In this research, annealing of compression-molded glass lenses
was studied. Two annealing experiments were performed. To
evaluate the impacts of the annealing experiments, residual
stresses and refractive index variation of the molded glass lens of
pre- and postannealing were compared. The residual stresses were
studied by a circular polariscope based on the property of birefrin-
gence of a glass lens [14–17]. The refractive index variation was
measured and reconstructed by an optical setup based on Mach–
Zehnder interferometer [7,18]. By comparing the experimental
results of pre- and postannealing experiments, it was discovered
that both residual stresses and refractive index variation were
reduced. In addition, curve deviations of the aspherical surface of
a glass lens after pre- and postannealing were also measured and
compared and determined to have experienced very minimal
changes.
2 Design of Experiments
2.1 Molded Aspherical Glass Lens. In this research,
compression-molded aspherical glass lenses were studied to inves-
tigate the impact of the annealing process. The aspherical glass
lenses were molded on a Toshiba commercial glass molding
machine (GMP-211V)[2]. Figure 1(a)shows a typical
1
Corresponding author.
Manuscript received January 3, 2013; final manuscript received June 19, 2013;
published online November 5, 2013. Assoc. Editor: Jack Zhou.
Journal of Manufacturing Science and Engineering FEBRUARY 2014, Vol. 136 / 011008-1
Copyright V
C2014 by ASME
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compression molding process, including mold and glass gob heat-
ing, molding, and cooling. Schott BK7 optical glass was chosen as
the lens material. BK7 glass is a borosilicate glass, a type of glass
in which its main glass-forming constituents are silica and boron
oxide, with low dispersion and relatively low refractive index.
The time–temperature history of the compression molding pro-
cess is shown in Fig. 1(b). Molding was conducted at 684 C.
Cooling of the glass lenses was performed in two steps, first at a
rate of 0.8 C/s to 520 C and then at a rate of 1.6 C/s to 200 C.
Once the temperature of the molds and glass lens was below
200 C, the glass lenses were taken out of the glass molding
machine and cooled naturally to room temperature.
The design of the aspherical glass lenses is shown in Fig. 2.
The aspherical surface is described by the following equation
Z¼Cx2
1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
11þKðÞC2x2
p(1)
where xis the coordinate of the optical surface, K(4.3666) is the
conic constant. Cis surface curvature (C¼1/R
1
,R
1
is the vertex
radius of the aspherical surface).
In Fig. 2, the section A-A was selected to evaluate the measured
residual stresses in a molded glass lens pre- and postannealing.
The results are discussed in Sec. 3.2.1.
2.2 Annealing Experiments. Annealing process of the
molded aspherical lenses was studied to investigate the reduction
of the level of residual stresses and refractive index variation. In
the annealing process, a molded glass lens was heated to a high
temperature and kept long enough to erase previous thermal his-
tory. This temperature is referred to as the soaking temperature.
When the molded glass lens reaches equilibrium at the soaking
temperature and subsequently undergoes slow cooling, the entire
glass lens will retain its homogeneity during cooling [10–13].
Table 1shows the thermal properties of BK7 glass [19]. At the
annealing point, the viscosity of glass is 10
13
P, the glass is soft
enough for the internal stresses to be released in a few minutes but
too hard to be deformed. At the strain point (511 C, g¼10
14.5
P),
the relaxation of stresses requires several hours.
Therefore, two types of annealing were designed: (1) a glass
lens was heated to 500 C and then soaked for 5 h. The furnace
was turned off after soaking for the lens to cool, and (2) a glass
lens was heated to 560 C, which is slightly higher than the glass’
transition temperature (T
g
¼557 C) and soaked for 10 min. The
furnace was turned off when the lens was cooled to 500 Cata
rate of 1 C/min. Figure 3shows the time–temperature history of
the experiments. The experiments were conducted in a commer-
cial furnace (Grieve, BF-12128-HT). Tested lenses were placed
on a ceramic plate with aspherical side facing down in the
furnace.
As shown in Fig. 3, in the annealing experiments, the cooling
rates became lower while the temperature of the furnace dropped
because there is no forced cooling in the furnace. The maximum
cooling rate is approximately 1.29 C/min after the furnace was
turned off at 500 C. Once the temperature of the furnace was
Fig. 1 (a) Schematic of a compression molding cycle and (b) time–temperature
history of the compression-molded glass lenses
Fig. 2 Aspherical lens: (a) design of aspherical lens (unit: mm) and (b) an aspherical glass lens
sample
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decreased to about 150 C, the glass lens was taken out of the fur-
nace and cooled to room temperature by natural cooling.
2.3 Residual Stresses Measurement. The measurement of
residual stresses is based on the property of birefringence when
polarized light passes through the glass lens. The property of bire-
fringence was analyzed by a circular polariscope. The schematic
of the circular polariscope is shown in Fig. 4(a). Figure 4(b)is the
experiment setup of the circular polariscope. In Fig. 4(a),S
denotes slow axis and Fdenotes fast axis of the quarter wave
plates and the glass lens. nand care the orientations of the slow
axis Sof the quarter-wave plates I and II, respectively. bis the ori-
entation of the analyzer. The orientation of the polarizer is set to
p/2. The polarizer and analyzer are two identical plane polarizers.
A red LED was used as the light source. To avoid refraction, the
lens under test was placed in a transparent box filled with refrac-
tive index matching liquid. The light beam carried with the lens’
information passing through the analyzer was captured by a CCD
camera.
In order to obtain the isoclinic angle uoff the slow axis Sand
optical retardation Dof the glass lens, a six-step left and right
phase shifting technique (PST) [20] was applied. The optical
arrangement for the six-step left and right PST is shown in
Table 2.
In the intensity equations shown in Table 2,I
0
is the amplitude
of light, and I
b
is the background light intensity. With the intensity
equations listed in Table 2, the parameters of isoclinic angle u
and optical retardation Dof the glass lens can be obtained as
u¼1
2arctan I5I3
I4I6
(2)
D¼arctan I5I3
ðÞsin 2uþI4I6
ðÞcos 2u
I1I2
(3)
Since the stress distribution is axisymmetric and the stresses are
low, the assumption of weak birefringence in the glass lens is
valid. This will allow the axial stress r
z
and shear stress s
rz
to be
determined directly from isoclinic angle uand retardation D
[14–17]. The relations between u,D, and the components of the
stress tensor along a light ray Lcan be expressed as [14]
Dk
2pcos 2u¼C0ðrzrx
ðÞdy (4)
Dk
2psin 2u¼2C0ðsxzdy (5)
where C
0
is the photoelastic constant, kis the wavelength of the
light ray. r
z
,s
xz
¼s
rz
/cosh, and r
x
are the components of the stress
tensor in the plane perpendicular to the light ray L(Fig. 5).
Fig. 3 Time–temperature history of the annealing experiments
Fig. 4 Circular polariscope for residual stresses measure-
ment: (a) schematic of the circular polariscope and (b) the
experiment setup of the circular polariscope
Table 2 Optical arrangement for the six-step left and right
phase shifting technique
ncb Intensity equation
1p/4 p/4 p/2 I1¼Ibþ1
2I0ð1þcos DÞ
2p/4 p/4 0 I2¼Ibþ1
2I0ð1cos DÞ
33p/4 0 0 I3¼Ibþ1
2I0ð1sin 2usin DÞ
43p/4 p/4 p/4 I4¼Ibþ1
2I0ð1þcos 2usin DÞ
5p/4 0 0 I5¼Ibþ1
2I0ð1þsin 2usin DÞ
6p/4 3p/4 p/4 I6¼Ibþ1
2I0ð1cos 2usin DÞ
Fig. 5 Cross-section of an axisymmetric lens
Table 1 Thermal properties of BK7 glass [19]
Thermal properties Viscosity (P) Value (C)
Strain point 10
14.5
511
Annealing point 10
13
557
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Using the equation of equilibrium
@rr
@rþrrrh
rþ@srz
@z¼0(6)
and the generalized sum rule
rhþrr¼rz2ðr
0
@srz
@zdr þC1(7)
other stress components, radial stress r
r
, and circumferential stress
r
h
, are determined [16,17]. C
1
is the integration constant deter-
mined by the boundary conditions at the surface of the specimen.
2.4 Refractive Index Variation Measurement. Refractive
index of the glass lens was measured by an optical setup based on
Mach–Zehnder interferometer [7,18]. The schematic of the optic
setup is shown in Fig. 6. He–Ne laser was used as the light source,
and the wavelength is k¼632.8 nm. The glass lens under test was
placed in a box filled with refractive index matching liquid. Fringe
patterns with refractive index distribution information of the lens
integrated at different angles were captured by a monochrome
CCD camera.
Based on the stress optic law and the amount of the residual
stresses inside the compression-molded glass lenses used in this
research, the differences among the refractive indices at different
directions of a point is at a level of 10
5
, which is considerably
less than the relative index between the glass lens and the optical
matching liquid (10
4
). To simplify the process of refractive
index reconstruction, the refractive index of glass lens was consid-
ered as a scalar.
The phase analysis of the interference fringe pattern was carried
out by a 2D Fourier transform technique [21] and a least-square
phase unwrapping method [22]. With the unwrapped phases,
three-dimensional (3D) refractive index distribution of the glass
lens can be reconstructed by the filtered back-projection method
[23]. Because of the axisymmetric property of the glass lens, the
refractive index was assumed to be axisymmetric as well. There-
fore, it is sufficient to reconstruct the refractive index in the glass
lens through interference fringe pattern in one direction.
The refractive index at a particular point in a molded glass lens
can be calculated by
nðx;y;zÞ¼pðx;y;zÞk
2pdþn(8)
where, p(x,y,z) is the reconstructed phase distribution at point (x,
y,z) through unwrapped phase by the filtered back-projection
method, dis the pixel size in the test object of the interferogram, n
is refractive index of the optical matching liquid.
3 Results and Discussion
3.1 Compression Molded Glass Lens. Figure 7shows the
measured residual stresses inside the molded aspherical lens in
Fig. 6 Optical setup for refractive index measurement
Fig. 7 Residual stresses of the molded glass lens in cylindrical coordinates: (a) axial stress r
z
,
(b) shear stress s
rz
,(c) radial stress r
r
, and (d) circumferential stress r
h
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cylindrical coordinates. The stress distribution displays half of the
cross section of the glass lens viewed in the normal direction. In
the stress distribution figures, negative value means compressive
stress and positive value means tensile stress. As glass lenses used
in this research were molded under the same parameters, residual
stresses inside the glass lenses are almost the same. Based on the
experimental results shown in Fig. 7, stresses r
z
and s
rz
show
larger magnitude at the edge of the lens compared to the stresses
in center. For r
r
and r
h
, the molded lens is mainly under tensile
stress in the center of the lens and compressive stress at the top
and bottom surfaces. In the measured residual stresses, some
errors were introduced at the edge and surface of the lens. How-
ever, the main purpose of this research is to investigate the glass
property change after experiments. The measured stresses are suf-
ficient to evaluate the changes.
3.2 Annealing Experiments.
3.2.1 Residual Stresses. Residual stresses in glass lenses after
annealing were also measured. Figure 8shows the measured resid-
ual stresses inside the glass lens after annealing experiment 1. The
stress distribution displays half of the cross section of the glass
lens viewed in the normal direction in order to compare with the
residual stresses preannealing experiment shown in Fig. 7. The
comparison between the residual stresses shown in Figs. 7and 8
shows that residual stresses inside the glass lens were significantly
released after annealing. For r
r
and r
h
, the maximum stresses
were decreased from 2.5 to 0.06 MPa at the center of the glass
lens. The stress distribution of the lens postannealing shows a sim-
ilar trend to the lens prior to the annealing experiment.
Figure 9shows the residual stresses measured after annealing
experiment 2. The stress distribution shows half of the cross sec-
tion of the glass lens viewed in the normal direction. Compared
with the residual stresses shown in Fig. 7, the residual stresses
inside the glass lens after experiment 2 in Fig. 9were also consid-
erably lower.
For a better demonstration of the stress decrease, residual
stresses in the middle section A-A (shown in Fig. 2) were com-
pared. Figures 10(a)and 10(b)show the stress components of
glass lens pre- and postannealing experiment 1 in the middle sec-
tion A-A, respectively.
Figure 11 shows a comparison of the stress components in the
middle section A-A of the glass lenses after annealing experiment
1 and annealing experiment 2.
Based on the results, these two annealing methods both can be
used to relieve the residual stresses trapped inside glass lenses.
Basically, lower cooling rates yield lower residual stresses. In
manufacturing, in order to get lower residual stresses, lower cool-
ing rate should be applied.
3.2.2 Refractive Index Variation. Figure 12 shows the aver-
age refractive index variation along the radial direction of aspheri-
cal glass lenses before and after the annealing experiments. As
shown in Fig. 12, the refractive index variation of the molded
glass lens decreased after annealing. The maximum reduction of
refractive index variation after annealing experiment 1 is less than
10
4
. After annealing experiment 2, the maximum reduction of
refractive index variation is about 4 10
4
, which is more than
half of the maximum variation before annealing.
According to the Lorentz–Lorenz equation, the relations
between refractive index nand density qcan be expressed as
[8,24,25]
dn
dq¼ðn21Þðn2þ2Þ
6nq(9)
Substituting volume Vfor the density q, refractive index change
Dncan be calculated from the volume change DVand the original
volume V
o
Dn¼ðn21Þðn2þ2Þ
6n
DV
VoþDV
(10)
Fig. 8 Residual stresses in the molded glass lens after experiment 1 in cylindrical coordinates:
(a) axial stress r
z
,(b) shear stress s
rz
,(c) radial stress r
r
, and (d) circumferential stress r
h
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After the annealing experiments, the volume change is described
by [25]
DV¼Vo3bh3bcþ12
Eðr11 þr22 þr33Þ
(11)
where, bh¼ÐT2
T1ahðTÞdT and bc¼ÐT1
T2acðTÞdT.a
h
and a
c
are
coefficient of thermal expansion (CTE) during heating and cool-
ing, respectively. The glass lens was heated from room tempera-
ture T
1
to soaking temperature T
2
and then cooled to T
1
.Eis
Young’s modulus, is Poisson’s ratio. r
11
,r
22
, and r
33
are the
changes of normal stresses along the axes.
In annealing experiment 1, the glass lens was heated to 500 C,
which is lower than its strain point (511 C). As the temperature
does not go through transition region, the CTE is the same as solid
CTE during heating and cooling. Therefore, stress relaxation is
the main factor of inducing the volume change. The expression of
volume change can be rewritten as
DV¼Vo
12
Eðr11 þr22 þr33Þ
(12)
According to the measured residual stresses shown in Figs. 7and
8, the maximum changes of normal stresses along the axes can be
determined, which are less than 3 MPa. Substitute the values of E
and into Eq. (12), the volume change can be calculated. For
BK7 glass lens, ¼0.206 and E¼82,500 MPa. Therefore,
DV¼610
5
V
o
. Substituting DVand n¼1.5148 into Eq. (10),
the maximum index change is 4 10
5
. However, measurement
error of the experiment is about 1 10
4
[18], which is larger
Fig. 10 Residual stresses at middle section: (a) stresses preannealing, and (b) stresses post-
annealing experiment 1
Fig. 9 Residual stresses in the molded glass lens after experiment 2 in cylindrical coordinates:
(a) axial stress r
z
,(b) shear stress s
rz
,(c) radial stress r
r
, and (d) circumferential stress r
h
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than the refractive index changes before and after experiment 1.
The difference of the refractive index variation of annealing
experiment 1 shown in Fig. 12 mainly comes from measurement
error.
In annealing experiment 2, the glass lens was heated to 560 C
which is higher than T
g
. CTE of a molded optical glass lens
depends not only on the temperature but also on the temperature
history in the molding process. When the glass goes through its
glass transition region, a
c
are different at different cooling rate.
As such, the volume change is induced by both CTE change and
stress relaxation. According to the measured stresses shown in
Sec. 3.2.1 and refractive index change calculation of experiment
1, the refractive index change caused by stress relaxation in
experiment 2 is similar to the refractive index change in experi-
ment 1. The maximum refractive index change caused by residual
stress relaxation is at a level of 10
5
. The maximum reduction of
refractive index variation is about 4 10
4
. Therefore, the refrac-
tive index change was mainly caused by CTE.
Based on the results, CTE has a significantly impact on the re-
fractive index in glass. In order to achieve a more homogenize re-
fractive index, annealing process starting above T
g
should be
considered. Again, slower cooling rates yield higher volume
change. Therefore, in order to get lower refractive index varia-
tions, a lower cooling rate should be used.
3.2.3 Curve Deviation. For aspherical glass lenses, geometry
of the aspherical surface is an important criterion to evaluate the
quality of glass lenses. In this study, the geometries of the aspheri-
cal surface in before and after experiments were measured on a
mechanical profilometer (Mitutoyo SV-3100). Figures 13 and 14
show the comparison of curve deviations of the aspherical surface
of glass lenses before and after annealing experiments 1 and 2,
respectively. The curve deviation was calculated by subtracting
the measured geometry to the desired geometry.
Based on the results shown in Figs. 13 and 14, the maximum
change of the curve is less than 1 lm. These results indicate that
the aspherical surfaces of the compression molding glass lenses
will retain their geometry after both annealing experiments
described in this study.
4 Conclusions
Residual stresses and refractive index are important criteria for
evaluating compression-molded glass lenses. The existence of re-
sidual stresses and refractive index variation in glass lenses could
impact the molded lenses’ optical performance. In this research,
two types of annealing experiments were carried out to study how
residual stresses and refractive index variation change when sub-
jected to different thermal treatments.
Fig. 11 A comparison of residual stresses at middle section of
glass lens after annealing experiments 1 and 2
Fig. 12 A comparison of the refractive index variation between
molded glass lenses before and after annealing experiments
Fig. 13 Comparison of curve deviations of aspherical surface
of compression-molded glass lens before and after annealing
experiment 1
Fig. 14 Comparison of the curve deviations of the aspherical
surface of the molded glass lens before and after annealing
experiment 2
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In order to evaluate the impacts of annealing, residual stresses
and refractive index variation in glass lenses from both pre- and
postannealing were compared. The residual stresses inside the
molded glass lenses were calculated based on the parameters
measured by a circular polariscope. An experiment setup based on
Mach–Zehnder interferometer was then used to measure and
reconstruct refractive index variation in glass lenses. In addition,
surface geometries of the glass lenses before and after the anneal-
ing experiments were also compared. The results showed that
annealing process could be used to reduce the residual stresses
and refractive index in molded glass lens, which maintain the sur-
face geometries.
From the experiments and analysis conducted in this research,
it was discovered that for the annealing experiment starting below
strain point, only residual stresses inside the molded lens are
reduced. Annealing experiment starting above T
g
, both residuals
stresses and refractive index can be reduced. Therefore, in order
to achieve a more homogenized refractive index, annealing
experiment starting above T
g
is recommended.
Based on the experimental results, a new compression molding
process may be proposed. During the cooling process in compres-
sion molding, a relatively high cooling rate is used in glass mold-
ing machine to maintain a proper production rate. An annealing
process can be added after molding to reduce the residual stresses
and refractive index variation in the molded glass lenses.
Future work would include study of the influence of annealing
of the compression molded glass lens by numerical simulation
and comparing the simulation results with experiment results to
better understand the annealing process.
Acknowledgment
The material is partially based on work supported by National
Science Foundation under Grants No. CMMI 0547311. Any opin-
ions, findings, and conclusions or recommendations expressed in
this material are those of the authors and do not necessarily reflect
the views of the National Science Foundation. The work is also
supported by National Natural Science Foundation of China
(Grant No. 51075381). Bo Tao acknowledges the financial support
from China Scholarship Council.
Nomenclature
K¼conic constant
C¼surface curvature, mm
1
R,r¼radius, mm
T
g
¼transition temperature, C
n¼angle of quarter wave plate I, radian
c¼angle of quarter wave plate II, radian
b¼angle of analyzer, radian
u¼isoclinic angle, radian
D¼retardation, radian
I¼light intensity
C
0
¼stress optic constant, MPa
1
r¼effective stress, MPa
s¼shear stress, MPa
k¼wavelength, nm
p¼phase, radian
d¼length, mm
n¼refractive index
q¼density, kg/m
3
V¼volume, mm
3
T¼temperature, C
a
h
¼coefficient of thermal expansion during heating, K
1
b
h
¼integral of coefficient of thermal expansion during heating
a
c
¼coefficient of thermal expansion during cooling, K
1
b
c
¼integral of coefficient of thermal expansion during cooling
E¼elastic modulus, MPa
¼Poisson’s ratio
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