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Using a Hexagonal Mirror for Varying Light Intensity
in the Measurement of Small-Angle Variation
Meng-Chang Hsieh 1 , Jiun-You Lin 2 and Chia-Ou Chang 1,3, *
1
2
3
*
Institute of Applied Mechanics, National Taiwan University, No. 1, Sec. 4 Roosevelt Road,
Taipei 10617, Taiwan; [email protected]
Department of Mechatronics Engineering, National Changhua University of Education, 2, Shi-Da Road,
Changhua 50074, Taiwan; [email protected]
College of Mechanical Engineering, Guangxi University, 100, University Road, Nanning 530004, China
Correspondence: [email protected]; Tel.: +886-2-3366-5626
Academic Editor: Thierry Bosch
Received: 16 June 2016; Accepted: 12 August 2016; Published: 16 August 2016
Abstract: Precision positioning and control are critical to industrial-use processing machines. In order
to have components fabricated with excellent precision, the measurement of small-angle variations
must be as accurate as possible. To achieve this goal, this study provides a new and simple optical
mechanism by varying light intensity. A He-Ne laser beam was passed through an attenuator and into
a beam splitter. The reflected light was used as an intensity reference for calibrating the measurement.
The transmitted light as a test light entered the optical mechanism hexagonal mirror, the optical
mechanism of which was created by us, and then it entered the power detector after four consecutive
reflections inside the mirror. When the hexagonal mirror was rotated by a small angle, the laser beam
was parallel shifted. Once the laser beam was shifted, the hitting area on the detector was changed;
it might be partially outside the sensing zone and would cause the variation of detection intensity. This
variation of light intensity can be employed to measure small-angle variations. The experimental results
demonstrate the feasibility of this method. The resolution and sensitivity are 3 × 10−40 and 4 mW/◦
in the angular range of 0.6◦ , respectively, and 9.3 × 10−50 and 13 mW/◦ in the angular range of 0.25◦ .
Keywords: hexagonal mirror; small angle; Gaussian beam; laser beam shifter
1. Introduction
In the industry of precise manufacturing, the techniques of optical small-angle measurement
have been extensively applied to the coordinates, alignment measurements of mechanical systems,
precision laser drilling [1], and positioning. Measuring the small angle is critical in precision-positioning
control application and mechanical calibration technology. Small-angle measurements in the
semiconductor industry have been aimed at setting up equipment easily, and miniaturizing and
attaining precise dimensions. Furthermore, some physical properties of materials also require precise
angle measurements, such as the angular displacement measurement of test samples twisted by
applied torques [2] for determining the relationship between shear stress and shear strain.
Small-angle variations can be measured using various methods such as heterodyne interferometry [3],
surface plasmon resonance [4], and image analysis [5]. However, the method of image analysis
needs off-line image processing to obtain the measurement results. The method of surface plasmon
resonance requires coating a thin metal film and the thermal effect [6] in the metal caused by the light
source will deteriorate the measurement accuracy. The method of heterodyne interferometry has both
polarization rotation and polarization mixing errors in the polarized components [7] which will cause
measurement errors. In addition, to accurately measure angles, the prisms in total internal reflection
and the mirrors in multiple light reflections [8–10] must demonstrate either superior right angles
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prisms in total internal reflection and the mirrors in multiple light reflections [8–10] must
demonstrate either superior right angles or parallelism; if the parallelism or right angle of a prism is
notparallelism;
perfect, it will
measurement
[7].a prism
There is
are
and measurement
displacement
or
if thecause
parallelism
or right errors
angle of
notexcellent
perfect, angular
it will cause
measurement
methods
[11–13]angular
but these
are methods
complicated.
To but
facilitate
errors
[7]. There
are excellent
and measurement
displacement structures
measurement
[11–13]
these
fabricating
components
with
high
precision
and
to
avoid
the
aforementioned
measurement
errors,
measurement structures are complicated. To facilitate fabricating components with high precision
this to
study
proposes
a new and simple
opticalerrors,
mechanism,
which
is a hexagonal
can
and
avoid
the aforementioned
measurement
this study
proposes
a new andmirror
simplethat
optical
parallel shift which
the light
and cause
variations
of light
intensity
at the
detector.
To cause
reducevariations
the effect
mechanism,
is path
a hexagonal
mirror
that can
parallel
shift the
light
path and
of light
unstable
sourceatintensity
on measurements,
the intensity
of thesource
test light
is divided
by that of the
of
intensity
the detector.
To reduce the effect
of unstable
intensity
on measurements,
reference
light.
system
will
present
highly
measurements
This
the
intensity
of theTherefore,
test light isthis
divided
by that
of the
reference
light.stable
Therefore,
this system [7,14].
will present
measurement
of
the
light
intensity
variation
can
be
utilized
to
evaluate
small-angle
variation.
The
highly stable measurements [7,14]. This measurement of the light intensity variation can be utilized
to
proposed
approach
provides
various
advantages,
such
as
a
simple
structure,
high
sensitivity,
and
evaluate small-angle variation. The proposed approach provides various advantages, such as a simple
high stability.
structure,
high sensitivity, and high stability.
2. Materials
Materialsand
andMethods
Methods
2.1.
2.1. Laser
Laser Beam
Beam Parallel
Parallel Shifter
Shifter
A
model
waswas
usedused
in thisinstudy.
mirrors
installed
four consecutive
A regular
regularhexagon
hexagon
model
this Four
study.
Four were
mirrors
wereoninstalled
on four
hexagonal
sidewalls
served
as
a
laser
beam
parallel
shifter
as
shown
in
Figure
1a.
When
a
laser
beam
consecutive hexagonal sidewalls served as a laser beam parallel shifter as shown in Figure 1a. When
hit
the first
mirror
and
reflected
four
1a, the(Figure
solid line),
the solid
reflected
a laser
beam
hit the
first
mirror off
andthrough
reflected
off mirrors
through(Figure
four mirrors
1a, the
line),light
the
off
the
un-rotated
hexagonal
mirror
was
defined
as
the
first
light.
When
the
hexagonal
mirror
rotated
reflected light off the un-rotated hexagonal mirror was defined as the first light. When the hexagonal
an
angle
∆θ, the
off the hexagonal
mirror
(Figure
1a, the
dash1a,
line)
the
mirror
rotated
an reflected
angle Δθ,light
the reflected
light off the
hexagonal
mirror
(Figure
thewas
dashcalled
line) was
second
light.
After
the After
second
the first the
lightfirst
path
of path
AB isofparallel
to the second
തതതത is parallel
called the
second
light.
themirror
secondreflection,
mirror reflection,
light
‫ܤܣ‬
to the
path
of
CD
as
shown
in
Figure
1a;
these
two
light
paths
are
also
parallel
after
the
fourth
mirror
തതതത
second path of ‫ ܦܥ‬as shown in Figure 1a; these two light paths are also parallel after the fourth
reflection
(the geometric
drawing
based on
the on
lawthe
of law
reflection,
that is,that
the is,
incident
angle is
equal
mirror reflection
(the geometric
drawing
based
of reflection,
the incident
angle
is
to
the to
reflective
angle,angle,
it directly
reveal
that the
lightlight
paths
are are
parallel
through
even
times
of
equal
the reflective
it directly
reveal
thattwo
the two
paths
parallel
through
even
times
reflections).
TheThe
shifted
distance
between
the first
second
is denoted
∆Z. Forby
hexagonal
of reflections).
shifted
distance
between
theand
first
and light
second
light is by
denoted
ΔZ. For
mirror,
∆Z is
increased,
the reflection
times
do. In addition,
the angle
variation
∆θ is,
hexagonal
mirror,
ΔZ isasincreased,
as the
reflection
times do.the
Inlarger
addition,
the larger
the angle
the
greaterΔθ
theis,shifted
distance
∆Z becomes,
as shown
in Figure
1a.
variation
the greater
the shifted
distance
ΔZ becomes,
as shown
in Figure 1a.
the
angular
variation.
(a) (a)
Hexagonal
mirror;
(b)
Figure 1. Parallel
Parallelshift
shiftlight
lightpaths
pathsbefore
beforeand
andafter
after
the
angular
variation.
Hexagonal
mirror;
Square
mirror.
(b)
Square
mirror.
The
toto
create
anan
angle
sensor
for for
measuring
the
The characteristics
characteristicsof
ofhexagonal
hexagonalmirror
mirrorwere
wereutilized
utilized
create
angle
sensor
measuring
rotation
angle
according
to to
thethe
distance
between
hexagonal
the
rotation
angle
according
distance
betweenthe
theparallel
paralleltwo
tworeflective
reflective lights.
lights. For
For hexagonal
mirror,
the
expression
relating
ΔZ
and
Δθ
is
given
by
Equation
(1),
mirror, the expression relating ∆Z and ∆θ is given by Equation (1),
Z  √
2 3 R sin(  ) ,
∆Z = 2 3Rsin(∆θ ),
(1)
(1)
which is derived in detail in the appendix. The R in Equation (1) is the circumradius and equal to the
which is derived in detail in the Supplementary Materials. The R in Equation (1) is the circumradius
side length of regular hexagon. Figure 1b shows the parallel shifted light path of square mirror before
and equal to the side length of regular hexagon. Figure 1b shows the parallel shifted light path of
and after the angle variation. The relationship between ΔZ and Δθ is derived as (see Appendix)
square mirror before and after the angle variation. The relationship between ∆Z and ∆θ is derived as
Zs  2 R sin( )
(see Supplementary Materials)
(2)
∆Zs = 2Rsin(∆θ )
(2)
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By the comparison between Equations (1) and (2), it is found that, for the same radius of
By
between
Equations
(1)
and
it is
is found
found
that,
for the
the same
same
radius
of
Bythe
thecomparison
comparison
between
Equations
(1)hexagonal
and (2),
(2), mirror
it
that,
for
radius
of
circumcircle
R
and same angle
variation
Δθ, the
has the
greater
shifted
distance
ΔZ
circumcircle
R
and
same
angle
variation
∆θ,
the
hexagonal
mirror
has
the
greater
shifted
distance
∆Z
circumcircle
R and
samemirror.
angle variation
Δθ,
the
hexagonal
mirrorprovides
has the greater
ΔZ
than
ΔZs of the
square
Therefore,
the
hexagonal
mirror
higher shifted
angulardistance
sensitivity
than
∆Zs
of
the
square
mirror.
Therefore,
the
hexagonal
mirror
provides
higher
angular
sensitivity
than
than
ΔZs
of
the
square
mirror.
Therefore,
the
hexagonal
mirror
provides
higher
angular
sensitivity
than the square mirror. At the same circumcircle size, the more the regular polygon number is, the
the
square
mirror.mirror.
At is,
theand
same
size, the
more
the
regular
polygon
number
is, theis,
better
than
the
At
the
same
circumcircle
size,
theis.
more
the regular
polygon
number
the
better
thesquare
sensitivity
thecircumcircle
less the
measurable
range
the
sensitivity
is,
and
the
less
the
measurable
range
is.
better
the sensitivity
is, between
and the less
thetwo
measurable
When
the distance
these
lights hadrange
been is.
measured, the variations of rotation angle
When
the
these
two
had been measured,
the
variations
ofrotation
rotation
angle
When
thedistance
distance
between
these
two lights
lights
variations
of
angle
would
be determined
bybetween
Equation
(1), where
the side-length
of regularthe
hexagon
is equal
to the radius
would
be
determined
by
Equation
(1),
where
the
side-length
of
regular
hexagon
is
equal
to
the
radius
would
be determined
byhere.
Equation
(1),
where the simulation
side-lengthresult using Equation
is (1)
equal
to the radius
R
and was
given 40 mm
Figure
2 shows
to demonstrate
Rthe
was
40
Figure
22 shows (Δθ)
the simulation
using Equation
Equation
(1)results
todemonstrate
demonstrate
Rand
and
wasgiven
givenbetween
40mm
mmhere.
here.
Figure
resultdistance
using
(1)
to
relationship
the angle
variation
and the shifted
(ΔZ). The
indicate
the
relationship
between
the
angle
variation
(∆θ)
and
the
shifted
distance
(∆Z).
The
results
indicate
the
relationship
between
the
angle
variation
(Δθ)
and
the
distance
(ΔZ).
The
results
indicate
that the angle and distance variation are relatively linear within ±3°. To prevent the last reflective
◦ . To prevent the last reflective
that
the
angle
and
distance
variation
are
relatively
linear
within
±
3
that the
angle
and distance
variationmirror,
are relatively
linear angle
withinwas
±3°.restricted
To prevent
the ±2°.
last reflective
point
being
outside
of the hexagonal
the rotational
within
point
2◦ .
pointbeing
beingoutside
outsideofofthe
thehexagonal
hexagonalmirror,
mirror,the
therotational
rotationalangle
angle was
was restricted
restricted within
within ±
±2°.
Figure 2. Relationship between Δθ and ΔZ.
Figure 2.
2. Relationship
Relationship between
and ΔZ.
Figure
between Δθ
∆θ and
∆Z.
2.2. Laser Parallel Shift and Light Intensity Variation
2.2.Laser
LaserParallel
ParallelShift
Shiftand
andLight
LightIntensity
Intensity Variation
2.2.
The hexagonal
mirror
is an optical Variation
mechanism for detecting the angle variation by measuring
The
hexagonal
mirror
is
an
optical
mechanism
for detecting
detecting
angle
measuring
the shifted
distance.mirror
As theislight
beam was
parallel shifted
due to the
motion by
of
mirror,
The hexagonal
an optical
mechanism
for
the angular
angle variation
variation
bythe
measuring
the
shifted
distance.
As
the
light
beam
was
parallel
shifted
due
to
the
angular
motion
of
the
the
hitting
point
would
advance
from
the
outer-casing
of
power
detector
(NOVA
II,
Sensor
size:
12.5
the shifted distance. As the light beam was parallel shifted due to the angular motion of themirror,
mirror,
the
hitting
point
would
advance
from
the
outer-casing
of
power
detector
(NOVA
II,
Sensor
size:
×
12.5
mm,
OPHIR,
Jerusalem,
Israel)
to
its
sensing
zone,
as
shown
in
Figure
3;
that
is,
the
hitting
point
the hitting point would advance from the outer-casing of power detector (NOVA II, Sensor 12.5
size:
× 12.5
mm,mm,
OPHIR,
Jerusalem,
Israel)
to
zone,
aspoint
shown
in Figure
3;2 that
the
hitting
point
1 sensing
at
the
outer-casing
boundary
marked
byits
○
to the
by
○
located
atis,
thethe
sensing
12.5
× 12.5
OPHIR,
Jerusalem,
Israel)
tomoved
its sensing
zone,
asmarked
shown
in Figure
3;is,that
hitting
1 laser
2 the
at theatAfter
outer-casing
marked
by of
○
moved
to the
point
marked
by
○
located
at the
sensing
2 , the boundary
zone.
○
Gaussian
waveform
was
completely
zone,
1beam
2 located
point
the outer-casing
boundary
marked
by moved
to
the
point inside
marked
bysensing
atthe
the
2 saturated.
zone. After
○
, the 2Gaussian
waveforminofintensity
laser beam
was completely
insideinthe
sensing
zone,
the
intensity
was
The
variation
is
particularly
prominent
the
vicinity
of
sensing zone. After , the Gaussian waveform of laser beam was completely inside the sensinglight
zone,
intensity
was saturated.
The variation
in intensity
is particularly
prominent
in the vicinity
of light
beam
crossing
boundary.
that
was used
as the sensing
range for
the
the
intensity
wasthe
saturated.
TheThus,
variation
insensitive
intensityrange
is particularly
prominent
in the vicinity
of light
beam crossing the boundary. Thus, that sensitive range was used as the sensing range for the
detection.
beam crossing the boundary. Thus, that sensitive range was used as the sensing range for the detection.
detection.
Figure
shifts corresponding
corresponding to
to light
light intensity
intensityvariations.
variations.
Figure 3.
3. Light
Light parallel
parallel shifts
Figure 3. Light parallel shifts corresponding to light intensity variations.
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2.3. The Definition of Laser Gaussian Distribution σ Value
TheThe
intensity
distribution
of theDistribution
laser can σ
beValue
considered a Gaussian distribution [15–17] having
2.3.
Definition
of Laser Gaussian
its general form expressed as Equation (3):
The intensity distribution of the laser can be considered a Gaussian distribution [15–17] having its
2 x2
general form expressed as Equation (3):

z r
z=
2
2
e
2
2  − 2x22
σ2 π
e
(3)
(3)
σ
In order to determine the value σ of laser Gaussian distribution, the laser light source was
Indisplaced
order to determine
thetovalue
σ of
Gaussian
distribution,
the laser
laser light
light source
source intensity
was
gradually
from 0 mm
4.2 mm
aslaser
shown
in Figure
3. When the
gradually
displaced
from
0
mm
to
4.2
mm
as
shown
in
Figure
3.
When
the
laser
light
source
intensity
is located at position ci (the central position of laser spot) its intensity is expressed by Equation (4) as
is located at position ci (the central position of laser spot) its intensity is expressed by Equation (4) as
2( x  ci ) 2
d 12.5 6.25
2d+12.5
I (ci ,  )  I 0 2 Z
2
I (ci , σ ) = I0 2 
σ π
d
 e
6.25
Z
d
6.25
e

2
2( x − c i )2
σ2
2 y2
e
e
2
dydx
2y2
dydx
σ2
,
,
(4)
(4)
−6.25
where Io represents the laser intensity and d is the distance from the origin of the coordinate to the
where Io represents
the laser
intensity
and dinisFigure
the distance
the and
origin
of the
coordinate
left boundary
of the sensing
zone
as shown
3. Thefrom
upper
lower
limits
of theto
y the
integral
left
boundary
of
the
sensing
zone
as
shown
in
Figure
3.
The
upper
and
lower
limits
of
the
y
integral
(±6.25 mm) are determined by the size of power detector. Equation (4) shows that the change of laser
6.25 mm)
by the size
detector. Equation (4) shows that the change of laser
spot(±
position
ci are
willdetermined
alter the intensity
I(cofi, power
σ).
spot position ci will alter the intensity I(ci , σ).
To evaluate the undetermined value σ, the numerical approximation conducts the principle of
To evaluate the undetermined value σ, the numerical approximation conducts the principle of
minimizing the square root of sum of squared difference between the experimental measurement
minimizing the square root of sum of squared difference between the experimental measurement
intensity
(mi)(m
and
theoretical
one
i, σ), which is expressed by Equation (5) as
intensity
) and
theoretical
oneI(c
I(c
, σ), which is expressed by Equation (5) as
i
i
v 8
u 8
Error ( ) u

( I (ci ,  )  mi ) 2
Error (σ ) = t ∑i 1( I (ci , σ ) − mi )2

(5)
(5)
i =1
The error function Equation (5) was calculated and plotted versus σ in Figure 4.
The error function Equation (5) was calculated and plotted versus σ in Figure 4.
Figure 4. The distribution of error between the experimental and theoretical values in different σ.
Figure 4. The distribution of error between the experimental and theoretical values in different σ.
σ was found to be 0.976 according to the minimum of error function shown in Figure 4.
TheThe
σ was
found to be 0.976 according to the minimum of error function shown in Figure 4. The
The theoretical intensity based on σ = 0.976 matches very well with the measured one as shown in
theoretical intensity based on σ = 0.976 matches very well with the measured one as shown in Figure 3.
Figure 3.
3. Experimental
Setup
3. Experimental
Setup
In the
proposed
experimental scheme,
scheme, the laser
Melles
Griot,
In the
proposed
experimental
laser light
lightsource
source(25LHP925-249,
(25LHP925-249,
Melles
Griot,
Carlsbad,
USA)
fitted
with
an attenuator
as shown
in Figure
5. When
rotary
stage
Carlsbad,
CA,CA,
USA)
waswas
fitted
with
an attenuator
as shown
in Figure
5. When
thethe
rotary
stage
(SGSP60YAW, Opto Sigma, Santa Ana, CA, USA) was rotated in a small angle and the laser beam was
shifted, the hitting area on the detector was changed and might be partially outside the sensing zone,
it would cause the variation of detection intensity, as shown in Figures 3 and 5. To avoid
measurement errors caused by the inclined hexagonal mirror and the unstable laser intensity, the
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(SGSP-60YAW, Opto Sigma, Santa Ana, CA, USA) was rotated in a small angle and the laser beam was
shifted, the hitting area on the detector was changed and might be partially outside the sensing zone,
it
would
cause
the
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1301variation of detection intensity, as shown in Figures 3 and 5. To avoid measurement
5 of
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errors caused by the inclined hexagonal mirror and the unstable laser intensity, the position calibration
position
calibration
the
mirror
was
performed
and
a beam
splitter
(BS)
was
used
split
athe
reference
of
the mirror
was performed
and was
a beam
splitter (BS)
was
used
to split
a reference
light
from
laser,
position
calibration
ofof
the
mirror
performed
and
a beam
splitter
(BS)
was
used
toto
split
a reference
light
from
the
laser,
respectively.
The
reference
light
intensity
(Ir)
detected
by
the
reference
power
respectively.
The
reference
light intensity
(Ir) is detected
by the reference
powerby
detector
(PDr) and
the
light
from
the
laser,
respectively.
The
reference
light
intensity
(Ir)
is is
detected
the
reference
power
detector
(PDr)
and
the
test
light
intensity
(It)
detected
the
test
power
detector
(PDt)
(for
intensity
test
light (PDr)
intensity
(It)
istest
detected
by
the test
power
detector
(PDt)
(for
intensity
stability
the
resolution
detector
and
the
light
intensity
(It)
is is
detected
byby
the
test
power
detector
(PDt)
(for
intensity
stability
the
resolution
the
power
was
chosen
to
1and
µW
this
study).
Both
are
of
the power
detector
was
chosen
to bedetector
1detector
µW inwas
this
study).
Both
Irin
are
error-prone
due
toand
a noise
stability
the
resolution
ofof
the
power
chosen
to
bebe
1ItµW
in
this
study).
Both
It It
and
Ir Ir
are
error-prone
duetoto
noise
andunstable
unstable
source(i.e.,
(i.e.,variations
variations
laserof
intensity
relative
the
and
unstable due
source
(i.e.,
variations
in
laser intensity
relative
to the duration
usage), relative
the
expression
error-prone
a anoise
and
source
ininlaser
intensity
totothe
duration
usage),
the
expression
It/Ir
would
be
used
instead
the
purpose
of
noise-elimination.
of
It/Ir would
be used
instead
for the
purpose
of be
noise-elimination.
Therefore,
thisofsystem
will present
duration
ofof
usage),
the
expression
ofof
It/Ir
would
used
instead
forfor
the
purpose
noise-elimination.
Therefore,
this
system
will
present
highly
stable
measurements.
highly
stable
measurements.
Therefore,
this
system
will
present
highly
stable
measurements.
Figure
Experimental
architecture.
Figure
5. 5.
Experimental
architecture.
Figure
5.
Experimental
architecture.
Results
and
Discussions
4. 4.
Results
and
Discussions
4.
Results
and
Discussions
4.1.
Experimental
Results
4.1.
Experimental
Results
This
study
proposed
a method
measuring
small-angle
variations
detecting
variations
proposed
method
forfor
measuring
small-angle
variations
variations
This
study
proposed
aa method
for
measuring
small-angle
variations
byby
detecting
thethe
variations
light
intensity.
The
rotation
the
hexagonal
mirror
a small
angle
causes
the
laser
beam
intensity.
Δθ
causes
the
laser
beam
toto
ofof
light
intensity.
The
rotation
ofof
the
hexagonal
mirror
byby
a small
angle
∆θΔθ
causes
the
laser
beam
to
parallel
shifted
ΔZ
(given
in
Equation
(1)),
and
this
affects
the
intensity
(given
in
Equation
(4))
bebe
parallel
shifted
byby
ΔZ
(given
in
Equation
(1)),
and
this
affects
the
intensity
(given
in
Equation
(4))
∆Z
the
light
entering
the
PDt;
therefore,
the
rotation
angle
Δθ
can
be
determined
the
variations
light
entering
the
PDt;
therefore,
the
rotation
angle
Δθ
can
bebe
determined
byby
the
variations
ofof
ofof
the
entering
the
PDt;
therefore,
the
rotation
angle
∆θ
can
determined
by
the
variations
the
intensity.
Figure
showsthe
theexperimental
experimentalresults
resultsof
thetest
testlight
light
intensity
(It).
Considerable
the
intensity.
Figure
(It).
Considerable
of
the
intensity.
Figure6 66shows
shows
the
experimental
results
ofofthe
the
test
intensity
(It).
Considerable
◦ and
◦ , and
variation
(high
sensitivity)
light
intensity
observed
between
0.6°
and
1.2°,
and
there
especially
variation
(high
sensitivity)
inin
light
intensity
is is
observed
between
0.6°
and
1.2°,
variation
(high
sensitivity)
in
light
intensity
is
observed
between
0.6
1.2
there
is is
especially
◦
◦
good
linearity
the
range
0.75°
and
1.1°;
therefore,
the
ranges
are
used
measuring
small-angle
good
linearity
inin
the
range
0.75°
1.1°;
the
ranges
good
linearity
in
the
range
ofof
0.75
and
1.1
; therefore,
are
used
forfor
measuring
small-angle
variations.
Figure
6
shows
that
the
theoretical
results
are
consistent
with
the
measured
ones.
variations.
Figure
6
shows
that
the
theoretical
results
are
consistent
with
the
measured
ones.
variations. Figure 6 shows that the theoretical results are consistent with the measured ones.
◦ )).
Figure
6.
Intensity
versus
angle
variation
(Power
(mW)
– Δθ
(°)).
Figure
6.6. Intensity
versus
angle
variation
(Power
(mW)
−
∆θ
((°)).
Figure
Intensity
versus
angle
variation
(Power
(mW)
– Δθ
4.2.
The
Measurement
Linearity
and
Reference
Light
Correction
4.2.
The
Measurement
of of
Linearity
and
Reference
Light
Correction
Figure7 7depicts
depictsthe
therelationship
relationshipbetween
betweenthe
theangle
anglevariation
variationand
andthe
theintensity
intensityininthe
thehighly
highly
Figure
sensitive
region
Figure
The
highly
sensitive
region
between
0.75°
and
1.1°
was
adopted.
Figure
sensitive
region
ofof
Figure
6. 6.
The
highly
sensitive
region
between
0.75°
and
1.1°
was
adopted.
InIn
Figure
7, 7,
2
2
a curve
the
linear
regression
showed
the
linearity
the
measurement
was
high
= 0.9987).
a curve
ofof
the
linear
regression
showed
the
linearity
ofof
the
measurement
was
high
(R(R= 0.9987).
Sensors 2016, 16, 1301
6 of 9
4.2. The Measurement of Linearity and Reference Light Correction
Figure 7 depicts the relationship between the angle variation and the intensity in the highly
sensitive region of Figure 6. The highly sensitive region between 0.75◦ and 1.1◦ was adopted. In Figure 7,
a curve of 16,
the1301
linear regression showed the linearity of the measurement was high (R2 = 0.9987). 6 of 9
Sensors2016,
Sensors2016, 16, 1301
6 of 9
Figure
versus angle
angle variation.
variation.
Figure 7.
7. The
The linear
linear regression
regression of
of intensity
intensity versus
versus
angle
variation.
This
This reference
reference correction
correction system
system (Figure
(Figure 5)
5) enables
enables the
the simultaneous
simultaneous detection
detection of
of the
the test
test and
and
This
reference
correction
system
(Figure
5)
enables
the
simultaneous
detection
of
the
test
and
reference
lights’
intensities.
The
correction
is
done
by
dividing
the
test
light’s
intensity
by
the
reference
lights’
intensities.
The
correction
is
done
by
dividing
the
test
light’s
intensity
by
the
reference lights’ intensities. The correction is done by dividing the test light’s intensity by the reference
reference
light’s
intensity.
The
left
vertical
axis
of
Figure
8
is
the
value
of
It/Ir.
The
right
vertical
axis
is
reference
light’s The
intensity.
The left
vertical
axis 8ofisFigure
8 is of
theIt/Ir.
valueThe
of It/Ir.
rightaxis
vertical
is
light’s
intensity.
left vertical
axis
of Figure
the value
rightThe
vertical
is theaxis
ratio
the
ratio
It/Ir
multiplied
by
the
stable-state
intensity
of
the
reference
light
(9
mW
was
the
stable-state
the
ratio
It/Ir
multiplied
by
the
stable-state
intensity
of
the
reference
light
(9
mW
was
the
stable-state
It/Ir multiplied by the stable-state intensity of the reference light (9 mW was the stable-state intensity
intensity
in
which
is
for
clarity
of
physical
quantity.
Figure
88 shows
intensity
in this
this experiment),
experiment),
which for
is added
added
for the
the
clarity
of the
thequantity.
physicalFigure
quantity.
Figurethat,
shows
in
this experiment),
which is added
the clarity
of the
physical
8 shows
after
2 = 0.9985).
that,
after
the
reference
correction,
there
was
still
good
linearity
(the
linear
regression,
R
2
that,reference
after the correction,
reference correction,
was still
good(the
linearity
linear regression,
R = 0.9985).
the
there was there
still good
linearity
linear(the
regression,
R2 = 0.9985).
Figure 8. Test light intensity divided by the reference light intensity for determining the small-angle
Figure 8. Test light intensity divided by the reference light intensity for determining the small-angle
regression curve.
variation
variation and
and plotted
plotted against
against the
the linear
linear regression curve.
4.3.
The
Sensitivity
and
4.3. The
The Sensitivity
Sensitivity and
and Resolution
Resolution
◦ in the
◦
The
is
superior
to
(mW/°)
∆power/
∆θ) of
architecture
0.6
The sensitivity
sensitivity (S
(S == Δpower/Δθ)
Δpower/Δθ)
of this
this architecture
architectureis
issuperior
superiorto
to444(mW/
(mW/°)) in
in the
the range
range of
of 0.6°
0.6° to
to
◦
◦ inthe
1.2°
1.2
(mW/
therange
rangeof
of0.75°
0.75◦to
to1.0°
1.0◦as
asshown
shownin
inFigure
Figure9.
9.If
Ifthe
thelaser
laserhas
hasgood
good intensity
intensity
1.2° and
and to
to 13
13 (mW/°)
(mW/°))in
in the
range
of
0.75°
to
1.0°
as
shown
in
Figure
9.
If
the
laser
has
good
intensity
and
and the
the detector
detector is
is of
of aa high
high resolution,
resolution, the
the sensitivity
sensitivity and
and resolution
resolution for
for the
the measurement
measurement of
of angle
angle
variations
can
be
improved.
The
relationship
between
Δθ
and
Δpower
can
be
expressed
as
follows:
variations can be improved. The relationship between Δθ and Δpower can be expressed as follows:



  
power
power

 power
power(()) 
(6)
(6)
Sensors 2016, 16, 1301
7 of 9
and the detector is of a high resolution, the sensitivity and resolution for the measurement of angle
variations can be improved. The relationship between ∆θ and ∆power can be expressed as follows:
∆θ =
Sensors2016, 16, 1301
Sensors2016, 16, 1301
∂θ
∂power (θ )
∆power
(6)
7 of 9
7 of 9
As revealed in Equation (6), ∆θ can be estimated by measuring ∆power. In addition, the resolution
of this system [14,18,19] can be defined as power
1
err
powererr
 err 
  
power
powererr 1
(7)



(7)
err ∆power
err

power
S
1S
err 
power
∆θ = ∆powererr
∆θerr = (7)
∆power S
The resolution
resolution of
of the
the power
power detector
detector in
in this
this study
study is
is 11 µW.
µW. The
The maximum
maximum and
and minimum
minimum
The
The
resolution
ofthe
thesignal
powererror
detector
in
this
study
is1.4
1 µW.
The
maximum
and
minimum
standardof the
standard
deviations
of
are
2.7
µW
and
nW.
The
average
standard
deviation
standard deviations of the signal error are 2.7 µW and 1.4 nW. The average standard deviation of the
−4°
deviations
of the
signal
are 2.7levels
µW and
nW.found
The average
standard
of the to
signal
signal
error is
is 1.2
1.2
µW.
The error
resolution
Δθ1.4
err are
from Figure
10 deviation
to be superior
10−4
signal
error
µW.
The
resolution
levels Δθ
err are found from Figure 10 to be superior to
33 ×× 10
°
−
40
error
is 1.2from
µW. The
resolution
levels
∆θ10
found
from Figure
10 to to
be 1.0°,
superior
to 3 × 10 They
in can
err−50are
in
the
ranges
0.6°
to
1.2°
and
9.3
×
in
the
ranges
from
0.75°
respectively.
−50
in the
1.2°
and9.3
9.3××10
10− 50 in
the ranges
rangesfrom
from0.75
0.75°
respectively.
◦ and
◦ to to
◦ , respectively.
theranges
rangesfrom
from 0.6°
0.6◦ to
to 1.2
in the
1.01.0°,
TheyThey
can can
be
evaluated
by
the
reciprocal
of
the
sensitivity
(S)
(Figure
9)
and
Equation
(7)
[19,20]
at
the
average
be evaluated
by the
reciprocal
ofofthe
(Figure9)9)and
andEquation
Equation
[19,20]
at average
the average
be evaluated
by the
reciprocal
thesensitivity
sensitivity (S)
(S) (Figure
(7)(7)
[19,20]
at the
standard deviation
deviation of
of the
the signal
signal error
error Δpower
Δpowererr
err = 1.2 µW.
standard
= 1.2 µW.
standard deviation of the signal error ∆powererr = 1.2 µW.
Figure 9.
9. The sensitivity
sensitivity of experimental
experimental architecture
architecture (σ
(σ is
is 0.976).
0.976).
Figure
Figure The
9. The sensitivityof
of experimental architecture
(σ is 0.976).
Figure
The
resolutionof
of experimental
experimental architecture
(σ (σ
is 0.976).
Figure
10.10.
The
resolution
architecture
is 0.976).
0.976).
Figure
10.
The
resolution
of experimental
architecture
(σ is
5. Conclusions
Conclusions
5.
The proposed
proposed hexagonal
hexagonal mirror
mirror architecture
architecture can
can be
be used
used to
to measure
measure small-angle
small-angle variations
variations in
in aa
The
new and
and simple
simple method.
method. Through
Through the
the observation
observation of
of the
the light
light intensity
intensity variation,
variation, the
the angle
angle variation
variation
new
Sensors 2016, 16, 1301
8 of 9
5. Conclusions
The proposed hexagonal mirror architecture can be used to measure small-angle variations in
a new and simple method. Through the observation of the light intensity variation, the angle variation
can be determined. The proposed design can be applied to detect reference light intensity and to
eliminate the factors caused by the laser intensity variation, which would affect the precision of the
measurement. The proposed structure is easy to assemble, low-cost, and highly stable. Based on these
advantages, it can be effectively used to measure small-angle variations. The resolution and sensitivity
are 3 × 10−40 and 4 mW/◦ in the angle interval of 0.6◦ , and 9.3 × 10−50 and 13 mW/◦ in the interval
of 0.25◦ , respectively.
Supplementary Materials: The supplementary materials are available online at http://www.mdpi.com/14248220/16/8/1301/s1, Hexagonal Mirror, Square Mirror.
Acknowledgments: The authors thank Pin-ruei Huang, Yi-Chi Wang and Jiun-Hong Lin for their knowledge and
mold manufacturing assistance on the optical system. The authors would like to thank the Ministry of Science
and Technology of R.O.C., for financially supporting this research under contract no. MOST 104-2221-E-002-017.
Author Contributions: Meng-Chang Hsieh designed the experiments; Meng-Chang Hsieh performed the
experiments; Meng-Chang Hsieh and Jiun-You Lin analyzed the data; Chia-Ou Chang is the thesis advisor.
All the three authors were engaged in writing this paper.
Conflicts of Interest: The authors declare no conflict of interest.
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© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC-BY) license (http://creativecommons.org/licenses/by/4.0/).