hv

photonics

Article
A Straightness Error Compensation System for Topography
Measurement Based on Thin Film Interferometry
Hang Su 1 , Ruifang Ye 1, * , Fang Cheng 1,2, *, Changcai Cui 3 and Qing Yu 1

1 College of Mechanical Engineering and Automation, Huaqiao University, Xiamen 361021, China;
18013080058@stu.hqu.edu.cn (H.S.); yuqing@hqu.edu.cn (Q.Y.)
2 Advanced Remanufacturing and Technology Centre (Agency for Science, Technology and Research),
Singapore 637143, Singapore
3 Institute of Manufacturing Technology, Huaqiao University, Xiamen 361021, China; cuichc@hqu.edu.cn
* Correspondence: yrf2010@hqu.edu.cn (R.Y.); chengf@artc.a-star.edu.sg (F.C.)

Abstract: Straightness error compensation is a critical process for high-accuracy topography mea-
surement. In this paper, a straightness measurement system was presented based on the principle of
fringe interferometry. This system consisted of a moving optical flat and a stationary prism placed
close to each other. With a properly aligned incident light beam, the air wedge between the optical
flat and the prism would generate the interferogram, which was captured by a digital camera. When
the optical flat was moving with the motion stage, the variation in air wedge thickness due to the
imperfect straightness of the guideway would lead to a phase shift of the interferogram. The phase
shift could be calculated, and the air wedge thickness could be measured accordingly using the image
processing algorithm developed in-house. This air wedge thickness was directly correlated with
the straightness of the motion stage. A commercial confocal sensor was employed as the reference
system. Experimental results showed that the repeatability of the proposed film interferometer



represented by σ was within 25 nm. The measurement deviation between the film interferometer and

 the reference confocal sensor was within ±0.1 µm. Compared with other interferometric straightness
Citation: Su, H.; Ye, R.; Cheng, F.; measurement technologies, the presented methodology was featured by a simplified design and good
Cui, C.; Yu, Q. A Straightness Error environment robustness. The presented system could potentially be able to measure straightness
Compensation System for in both linear and angular values, and the main focus was to analyze its linear value measurement
Topography Measurement Based on capability.
Thin Film Interferometry. Photonics
2021, 8, 149. https://doi.org/
Keywords: straightness measurement; film interferometry; image processing; phase shift; robustness
10.3390/photonics8050149

Received: 19 April 2021

Accepted: 28 April 2021
1. Introduction
Published: 30 April 2021
Precision positioning is a key technological enabler for the advanced manufactur-
Publisher’s Note: MDPI stays neutral ing [1–4] and instrumentation industry [5]. For a precision positioning system, straightness
with regard to jurisdictional claims in is a very important parameter to assess performance [6]. Especially in the area of topog-
published maps and institutional affil- raphy measurement, the straightness error of the positioners will significantly affect the
iations. measurement accuracy [7,8]. How to quantify and compensate the straightness error has
become an important research topic in this area [9,10]. In practice, the straightness error
can be expressed through linear values (in microns or nanometers) or angular values (in
arcseconds or microradians) [11–14]. Different approaches have been published to quantify
Copyright: © 2021 by the authors.
the straightness in either way.
Licensee MDPI, Basel, Switzerland. There are three types of straightness measurement principles: mechanical-datum-
This article is an open access article based (Type 1) [15,16], collimated-light-based (Type 2) [17,18], and interferometry-based
distributed under the terms and (Type 3) [19–21].
conditions of the Creative Commons Type 1 is a traditional methodology for measuring linear values in which a displace-
Attribution (CC BY) license (https:// ment probe is usually used to trace the off-axis offsets of a reference line or plane. In
creativecommons.org/licenses/by/ Ref. [15], a taut nylon fishing wire was the known reference, and the wire offset was
4.0/).

Photonics 2021, 8, 149. https://doi.org/10.3390/photonics8050149 https://www.mdpi.com/journal/photonics

Photonics 2021, 8, 149 2 of 18

detected by a slotted optical sensor. In Ref. [16], a focused beam formed the displace-
ment probe to trace the reference plane of a reflective wafer surface. The accuracy of this
methodology relied on the accuracy of the probe and the quality of the reference line
or plane.
Represented by autocollimators, Type 2 is a popular way of quantifying motion errors
(mostly angular values) of machine tools and other long-travel systems based on the
autocollimation principle and reflection principle. In Ref. [17], a miniature three-degree-of-
freedom laser measurement system, including a miniature autocollimator kit, was proposed
to measure the straightness of a precision positioning stage. In Ref. [18], a three-axis angular
motion error simultaneous detection system composed of two autocollimation units was
presented. In this type of system, positioning sensitive detectors (PSDs) were normally
used to detect the spot movements, which were correlated with the straightness of the
motion stages. Therefore, the accuracy of this methodology was limited by the resolution
of the position sensors.
Type 3 based on optical interferometry is suitable for high accuracy measurements. The
straightness errors can be obtained from the change in optical path difference. In Ref. [19],
a straightness measurement system based on a 2D encoder was proposed. The ±1st
order diffracted beams formed the interferogram to measure the displacement correlated
with straightness. In Ref. [20], a multi-probe measurement system was equipped with a
micro-coordinate measuring machine, in which two laser interferometers were used to
separate the angular motion error. In Ref. [21], a six-degree-of-freedom laser straightness
interferometer system was proposed to obtain the angular errors when the stage was
moving.
Although Type 3 is a high-accuracy solution, it is quite sensitive to environmental
factors such as temperature variation and airflow in most applications. It may require
a strict assembly process of the optics [22], additional preprocessing circuits [23,24], and
customized signal processing algorithms [25]. These disadvantages limit its applications,
especially when an in-situ measurement is required. An interferometric solution with high
robustness, therefore, became one of the motivations of the presented work.
Another motivation of the presented work was to develop an interferometric system,
which was able to quantify and compensate the motion error from the bottom of the
3D surface topography measurement instrument [26]. Traditional laser interferometric
systems, based on the Michelson principle, are usually used for measuring one-dimensional
angular motion errors (yaw or pitch) during the movement of the stage [27,28], as shown
in Figure 1a. In order to measure multi-degree-of-freedom motion errors at the same
time, more than one interferometric module is required. Furthermore, yaw has no effect
on the height measurement, so it is less critical in topography measurement. The film
interferometer, presented in this paper, is capable of measuring straightness errors in the
vertical direction and angular values (yaw and roll) simultaneously, which directly affect
the height measurement, as shown in Figure 1b. If a traditional Michelson interferometer
was built to meet this requirement, a flawless reflective mirror would be needed to cover the
entire measurement area. Such large mirrors with perfect flatness and surface quality are
not practically available. In the presented work, optical flats, with very affordable flatness
references, were used to generate the interferogram. Further, image processing based on
areal sampling was able to effectively minimize the impact of surface imperfections.
Photonics 2021, 8, 149 3 of 18
Photonics 2021, 8, x FOR PEER REVIEW 3 of 18

(a)

(b)
Figure 1.
Figure 1. Comparison
Comparisonbetween
betweenthe
thetraditional laser
traditional interferometer
laser andand
interferometer the the
filmfilm
interferometer: (a)
interferometer:
degree
(a) of freedom
degree measured
of freedom by the
measured bytraditional laser interferometer;
the traditional (b) degree
laser interferometer; of freedom
(b) degree meas-
of freedom
ured by the film interferometer.
measured by the film interferometer.

2. Principle of Film
In summary, theInterference
motivation Module
of this study was to develop a straightness measure-
mentIn system, which was not only able
order to provide detailed informationto verify on
a motion system but work,
the development also tothe
improve the
principle,
accuracy for topography measurement with reasonable cost and respectable
calibration process, and data analysis were discussed based on a standalone experimental robustness.
Being
setup,applied
as shown in in
anFigure
in-house-developed
2. topography measurement system, the proposed
methodology was proven
The experimental by significantly
system improving
mainly included the flat
an optical measurement accuracy
and a right-angle [26].The
prism. In
this paper,
optical flat details on themounted
was a datum optical scheme, image linear
on a precision quality, implementation
stage under test andmethod,
formed an error
air
factors, and system robustness were disclosed.
wedge with a stationary right-angle prism. The angle of the air wedge was very small,
usually within 100″. With a proper incident direction, the light source was split into two
2. Principle of Film Interference Module
beams and reflected with the same propagation path approximately, and a film interfero-
gramIncould
orderbetoobserved
provide on detailed information
the screen. During theon the development
movement work, the
of the optical flat,principle,
the inter-
calibration process,
ferogram would andadata
show phaseanalysis were discussed
shift according based on avariation
to the thickness standalone experimental
of the air wedge,
setup,
whichas shown
was in Figure
directly 2. to the straightness error of the linear stage. Therefore, the
related
The experimental system
straightness error could be measured mainlybased
included
on theanphase
optical flatofand
shift the ainterferogram.
right-angle prism.
The optical flat was a datum mounted on a precision linear stage under test and formed
an air wedge with a stationary right-angle prism. The angle of the air wedge was very
small, usually within 100”. With a proper incident direction, the light source was split
into two beams and reflected with the same propagation path approximately, and a film
interferogram could be observed on the screen. During the movement of the optical flat,
the interferogram would show a phase shift according to the thickness variation of the air
wedge, which was directly related to the straightness error of the linear stage. Therefore,
the straightness error could be measured based on the phase shift of the interferogram.

Figure 2. Schematic drawing of the film interference module.

As shown in Figure 2, the optical path difference between the two interfering light
beams can be expressed as:
 wedge with a stationary right-angle prism. The angle of the air wedge was very small,
usually within 100″. With a proper incident direction, the light source was split into two
beams and reflected with the same propagation path approximately, and a film interfero-
gram could be observed on the screen. During the movement of the optical flat, the inter-
Photonics 2021, 8, 149 ferogram would show a phase shift according to the thickness variation of the air wedge, 4 of 18
which was directly related to the straightness error of the linear stage. Therefore, the
straightness error could be measured based on the phase shift of the interferogram.

Figure 2. Schematic drawing

Figure of the film
2. Schematic interference
drawing module.
of the film interference module.

As shown inAsFigure
shown2,inthe optical
Figure 2, path difference
the optical pathbetween thebetween
difference two interfering light
the two interfering light
beams can be expressed
beams can beas:
expressed as:

d = n1 (AB + BC + HI) − n2 (AE + FG) (1)

where n1 and n2 are the refractive indexes of the air and optical glass, respectively. Since
the angle and thickness of the air wedge were very small, AB was approximately equal
to BC:
AB = BC (2)
So Equation (1) can be modified as:

d = (2n1 AB − n2 AE) + (n1 HI − n2 FG) (3)

Set GH = a, HI and FG can be obtained through geometric relationships:

HI = asinθ1 (4)

FG = asinθ2 (5)
Based on the Snell’s laws [29]:

n1 sin θ1 = n2 sin θ2 (6)

n2 sin θ3 = n1 sin θ4 (7)

The right half of Equation (3) can be calculated:

n1 HI − n2 FG = n1 asinθ1 − n2 asinθ2 = n1 asinθ1 − n1 asinθ1 = 0 (8)

The distance between the two interference surfaces was set as BD = h, so the optical
path difference d can be expressed as:
 
2n1 h 2n1 h n1 n2 sin θ3 sin θ4
d = 2n1 AB − n2 AE,= − 2n2 ADsinθ3 ,= − 2n2 htanθ4 sin θ3 ,= 2h − . (9)
cos θ4 cos θ4 cos θ4 cos θ4

According to Equation (7), Equation (9) can be further simplified:

!
1 sin2 θ4
d = 2hn1 − = 2hn1 cos θ4 (10)
cos θ4 cos θ4
Photonics 2021, 8, 149 5 of 18

Equation (10) is the classic optical path difference formula for the parallel air wedge.
Substituting Equations (6) and (7) into Equation (10), which can be modified as:

d = 2hn
p1 cos θ4
= 2hqn1 2 − n2 2 sin2 θ3 ,
= 2h n1 2 − n2 2 sin2 π4 − θ2 ,


p (11)
= hr4n1 2 − 2n2 2 + 4n1 n2 sin θ1 cos θ2 ,
q 2
= h 4n1 2 − 2n2 2 + 4n1 n2 sin θ1 1 − nn1 2 sin2 θ1 .
2

Since the refractive indexes of the optical flat and the right-angle prism were the same
in the system, the half-wave rectification caused by reflection must be considered:
v s
u
λ u n1 2
d = +h 4n1 2 − 2n2 2 + 4n1 n2 sin θ1 1− sin2 θ1 (12)
t
2 n2 2

The relationship between the change in the optical path difference ∆d and the change
in distance ∆h (straightness error) can be expressed as:
v s
u
u n1 2
∆d = 4n1 2 − 2n2 2 + 4n1 n2 sin θ1 1− sin2 θ1 × ∆h (13)
t
n2 2

When the phase shift of the interferogram ∆ϕ was equal to 2π (one wave cycle), the
optical path difference would be changed by one wavelength. Hence, the phase shift ∆ϕ
was linearly correlated with the optical path ∆d:
v s
u
2π 2π u n1 2
∆ϕ = ∆d × = × 4n1 2 − 2n2 2 + 4n1 n2 sin θ1 1− sin2 θ1 × ∆h. (14)
t
λ λ n2 2

The conditional parameters, including wavelength λ, the refractive indexes n1 and

n2 , and the incident angle θ1 were independent with the phase shift ∆ϕ. Hence, the item
before ∆h can be substituted by a constant c:
v s
u
2π u n1 2
c= × 4n1 2 − 2n2 2 + 4n1 n2 sin θ1 1− sin2 θ1 (15)
t
λ n2 2

And Equation (14) can be modified as:

∆ϕ = c × ∆h. (16)

Equation (16) shows that the magnitude and direction of the straightness error ∆h can
be calculated by dividing the phase shift ∆ϕ of the interferogram by the linear coefficient c.

3. Analysis of Angular Motion Error

Straightness error can be expressed either in linear values (in microns or nanometers)
or angular values (in arcseconds or microradians). The present work was focused on
linear values. According to the analysis in the above section, c is irrelevant with the angle
of the air wedge affected by angular motion error. In order to verify the conclusion, a
simulation-based on ZEMAX was conducted.
In general, the angular motion error of a precision linear stage can be well controlled
within 60”, so the setting of θ to be 20”, 50”, and 80” was sufficient to simulate the impact
of the angular motion error. As shown in Figure 3, the angular motion error affected the
Photonics 2021, 8, 149 6 of 18

spacing and orientation of the interferogram. Figure 4 shows that the relationship between
the distance and phase shift remained consistent with a different set of angles. As shown in
Table 1, the linear coefficient c was independent of the preset angle. Therefore, the angular
motion error was a separate topic and did not affect the linear straightness measurement
Photonics 2021, 8, x FOR PEER REVIEW 6 of 18
method proposed in this paper, at least within the operational range.

Figure 3. Angular motion error.

Table 1. Calibration results of the linear coefficient c.

Order Δh (μm) Δφ (rad) c (rad/μm)

10.00 89.01 8.901
θ = 20″ 30.00 267.04 8.901
50.00 445.07 8.901
6.67 59.37 8.901
θ = 50″ 26.67 237.40 8.901
46.67 415.42 8.901
8.33 74.15 8.902
Figureθ3.=Angular
80″ motion 28.33
error. 252.16 8.901
Figure 3. Angular motion error.
48.33 430.17 8.901
Table 1. Calibration results of the linear coefficient c.

Order Δh (μm) Δφ (rad) c (rad/μm)

10.00 89.01 8.901
θ = 20″ 30.00 267.04 8.901
50.00 445.07 8.901
6.67 59.37 8.901
θ = 50″ 26.67 237.40 8.901
46.67 415.42 8.901
8.33 74.15 8.902
θ = 80″ 28.33 252.16 8.901
48.33 430.17 8.901

Figure
Figure 4.4.Simulation
Simulationresults
results
ofof the
the phase
phase shift
shift atat different
different angles.
angles.

Therefore, it could be concluded that, although the angular motion error varied in
the appearance of the interferogram, it did not affect the mathematical model.

Figure 4. Simulation results of the phase shift at different angles.

Therefore, it could be concluded that, although the angular motion error varied in
the appearance of the interferogram, it did not affect the mathematical model.
Photonics 2021, 8, 149 7 of 18

Table 1. Calibration results of the linear coefficient c.

Order ∆h (µm) ∆ϕ (rad) c (rad/µm)

10.00 89.01 8.901
θ = 20” 30.00 267.04 8.901
50.00 445.07 8.901
6.67 59.37 8.901
θ = 50” 26.67 237.40 8.901
46.67 415.42 8.901
8.33 74.15 8.902
θ = 80” 28.33 252.16 8.901
48.33 430.17 8.901

Photonics 2021, 8, x FOR PEER REVIEW 7 of 18

Therefore, it could be concluded that, although the angular motion error varied in the
appearance of the interferogram, it did not affect the mathematical model.

4.
4. Phase
Phase Calculation
Calculation Based
Based on
on Image
Image Processing
Processing
The
The direct mathematical calculationof
direct mathematical calculation ofthe
thephase shift∆ϕ
phaseshift Δφwaswaschallenging
challengingdue
dueto
tothe
the
imperfection the interferogram. In this study, the phase shift ∆ϕ
imperfection of the interferogram. In this study, the phase shift Δφ was determined byan
of was determined by an
image
image processing
processingalgorithm.
algorithm.

4.1.
4.1. Analysis
Analysis of
of the
the Cause
Cause of
of Fringe
Fringe Distortion
Distortion
Assuming
Assuming the incident light was
the incident light was reflected
reflected only
only once
once by
by each
each optical
optical surface,
surface, aa two-
two-
beam
beam interferogram would be generated. Figure 5a shows the ZEMAX simulationresults
interferogram would be generated. Figure 5a shows the ZEMAX simulation results
of the two-beam scenario. The grayscale variation showed an ideal sinusoidal pattern. In
of the two-beam scenario. The grayscale variation showed an ideal sinusoidal pattern. In
this ideal scenario, the phase information could be easily obtained by methods such as the
this ideal scenario, the phase information could be easily obtained by methods such as the
Fourier transform. However, in the actual experiments, the interferogram generated by the
Fourier transform. However, in the actual experiments, the interferogram generated by
actual optics showed multi-beam patterns. The incident light was reflected more than once
the actual optics showed multi-beam patterns. The incident light was reflected more than
between the two optical surfaces. When the multi-beam reflection was set as the initial
once between the two optical surfaces. When the multi-beam reflection was set as the ini-
condition for the ZEMAX simulation, the interferogram showed a significant skewness, as
tial condition for the ZEMAX simulation, the interferogram showed a significant skew-
shown in Figure 5b.
ness, as shown in Figure 5b.

(a)

(b)
Figure5.5. Interference
Figure Interferencephenomena:
phenomena:(a)
(a)two-beam
two-beaminterference
interference simulation;
simulation; (b)(b) multi-beam
multi-beam interfer-
interference
ence simulation.
simulation.

As discussed in the above paragraphs, the root cause of the waveform distortion was
the multi-reflection that occurred in the air wedge, as illustrated in Figure 6.
Photonics 2021, 8, 149 (b) 8 of 18
Figure 5. Interference phenomena: (a) two-beam interference simulation; (b) multi-beam interfer-
ence simulation.

As discussed in the above paragraphs, the root cause of the waveform distortion was
As discussed in the above paragraphs, the root cause of the waveform distortion was
the multi-reflection that occurred in the air wedge, as illustrated in Figure 6.
the multi-reflection that occurred in the air wedge, as illustrated in Figure 6.

Photonics 2021, 8, x FOR PEER REVIEW 8 of 18

Photonics 2021, 8, x FOR PEER REVIEW 8 of 18

Figure 6. Schematic diagram of the multi-beam interference.

Figure 6. Schematic diagram of the multi-beam interference.
The6.waveform
Figure distortion,
Schematic diagram due
of the to multi-reflection,
multi-beam interference.could be quantitatively expressed
by the The waveform
coefficient distortion,
of finesse F [30]:due to multi-reflection, could be quantitatively expressed
by the coefficient of finesse
The waveform distortion, F [30]:
due to multi-reflection, could be quantitatively expressed
4ρ
by the coefficient of finesse F [30]: F = 4ρ (17)
F(=
1 − ρ )2 (17)
1−4ρ 𝜌
F= (17)
where
whereρ ρis is
the interface
the interfacereflectance.
reflectance.According
According 1to
−toEquation
𝜌 Equation(17),
(17),a ahigher
highervalue
valueofofρ will
ρ will
result in a ahigher
result value
valueofof
F.F.
With the finesse F increasing, the stripes become finer, and
where inρ is higher
the interface With
reflectance. the finesse
According F to
increasing,
Equation the
(17),stripes become
a higher valuefiner, and
of ρ will
their edges
their edges become
become sharper.
sharper. This correlation was represented by wider bright stripes in
result in a higher value of F. This
Withcorrelation
the finesse was represented
F increasing, the by wider
stripes bright finer,
become stripes in
and
the interferogram generated by the ZEMAX simulation, as shown in Figure 7. The actual
the interferogram
their captured
edges become generated by the ZEMAX simulation, as shown in Figure 7. The actual
image from sharper. This correlation
the experimental systemwas wasrepresented by wider
consistent with bright stripes
the simulation andin
image
the captured from
interferogram the experimental
generated by the ZEMAX system was consistent
simulation, as shown with
in the simulation
Figure 7. The and
actual
mathematical analysis, as shown in Figure 8.
mathematical
image captured analysis,
from theas shown in Figure
experimental 8.
system was consistent with the simulation and
mathematical analysis, as shown in Figure 8.

Figure 7. Interferogram with different parameters.

Figure 7. Interferogram with different parameters.
Figure 7. Interferogram with different parameters.

Figure 8. Actual interferogram.

Figure 8. Actual interferogram.
Figure 8. Actual
4.2. Edge interferogram.
Extraction and Phase Calculation
In this
4.2. Edge study, interferograms
Extraction were captured continuously during the movement of
and Phase Calculation
the optical
In this study, interferograms werecalculated
flat. The phase values were captured in real-time. Due
continuously to the
during themulti-beam
movementin-of
terference,
the optical the
flat.fringes showed
The phase sharp
values edges.
were Therefore,
calculated edge detection
in real-time. Due totechnology was ap-
the multi-beam in-
plied to identify
terference, the interference
the fringes signal,
showed sharp as shown
edges. in Figure
Therefore, edge 9:
detection technology was ap-
 Photonics 2021, 8, 149 9 of 18

4.2. Edge Extraction and Phase Calculation

In this study, interferograms were captured continuously during the movement of
the optical flat. The phase values were calculated in real-time. Due to the multi-beam
tonics 2021, 8, x FOR PEER REVIEW interference, the fringes showed sharp edges. Therefore, edge detection9technology
of 18 was
applied to identify the interference signal, as shown in Figure 9:

Photonics 2021, 8, x FOR PEER REVIEW 9 of 18

Figure 9. Edge extraction based on the image processing technique.

Figure 9. Edge extraction based on the image processing technique.

The red dotStep

in Figure 9 represents
1. Central the location
area selection for straightness
to minimize the optical measurement. If thethe imaging
distortion from
Figure
fringe edge system. 9. Edge extraction based on the image processing technique.
is on the left side of the dot, the distance L is a positive number; otherwise, it
is negative. As shown
Step 2.inImage
Figure 10, assuming
enhancement tothat the phase
improve of thecontrast.
the image first interferogram φ1
is 0, and the phase Theof red
the dot
nth in Figure 9
interferogram represents
was φ n, the
thenlocation
Step 3. Hole filling to patch the holes induced by theofthe phasefor straightness measurement.
the (n + 1)th
binarization inter-
process. If the
ferogram φn+1fringe
can edge
be
Step 4. is on theas:left
expressed
Opening side of the
operation dot, thethe
to reduce distance
noise at L is
thea positive
edges andnumber;
separateotherwise,
edges ofit
is negative. As
neighboring fringes. shown in Figure 10, assuming that the phase of the first interferogram φ1
L L
is 0, Step
and the phase φof the= 2π
nth
5. Edge extraction to store × −
interferogram + φ
was . φn, then the phase of the (n
(18) + 1)th inter-
ferogram φn+1dot
caninbe expressed T edgeTinformation in the format of pixel arrays.
as:
The red Figure 9 represents the location for straightness measurement. If the
Transforming Equation
fringe edge is on (18) to obtain
the left side ofthe
thephase
dot, difference
L
the distance ΔφL Ln:is a positive number; otherwise,
φ 10,=assuming
2π × − the+phase φ . of the first interferogram (18)
it is negative. As shown in Figure L L T that T (19)
ϕ1 is 0,∆φand = the
φ phase − φ of= the
2π ×nth interferogram
− was ϕn , then the phase of the (n + 1)th
Transforming Equation T to obtain
(18) T the phase difference Δφn:
interferogram ϕn+1 can be expressed as:
 L L  (19)
∆φ = φ − φ = 2π ×Ln+1 −Ln
ϕn+1 = 2π × T − T + ϕn . (18)
Tn+1 Tn

(a) (b)
Figure 10. Phase calculation method: (a) edge of the nth interferogram; and (b) edge of the (n + 1)th
interferogram.
(a) (b)
5. Experimental Analysis
Figure
Figure 10.Phase
10. Phasecalculation
calculationmethod:
method:(a)
(a)edge
edgeof
ofthe
thenth interferogram; and
nth interferogram; and (b)
(b) edge
edge of
of the
the(n
(n++ 1)th
1)th
In orderinterferogram.
to verify the proposed method, the experimental tests were conducted with
interferogram.
the system shown in Figure 11. The key components with the specifications are listed in
Table 2.
5. Experimental Equation (18) to obtain the phase difference ∆ϕn :
TransformingAnalysis
In order
The optical flat to verify on
was mounted thethe
proposed method,
linear stage the experimental
as a geometric
 tests
reference
 andwere
formedconducted with
Ln+1 Ln
an air wedgethe system
with the stationary ∆ϕn =
shown in right-angle
Figure 11.ϕThe −key
prism.
n+1 ϕ components
The
n = 2π
two × with
mirrors − the
were specifications
used to adjust are listed in
(19)
Table 2. Tn+1 Tn
the incident light. The linear stage was driven by an MC600 controller (Zolix Instruments
Co., Ltd., Beijing,The optical
China). flat was
During themounted on the linear
stage movement, stage as a geometric
interferogram reference
images were cap- and formed
an air wedge with the stationary right-angle prism. The two mirrors
tured at 51 locations with an increment of 1 mm. The confocal sensor was used as a refer- were used to adjust
the incident
ence for calibration light. The linear
and comparison. stage
Since onlywas driven
about 5% ofbythe
an measurement
MC600 controllerrange (Zolix
was Instruments
Co.,
used, it could be Ltd., Beijing,
considered China).
that During theerror
the nonlinearity stageinmovement, interferogram
this range was images were cap-
insignificant.
tured at 51 locations with an increment of 1 mm. The confocal sensor was used as a refer-
Photonics 2021, 8, 149 10 of 18

5. Experimental Analysis
In order to verify the proposed method, the experimental tests were conducted with
the
Photonics 2021, 8, x FOR PEER REVIEW system shown in Figure 11. The key components with the specifications are listed
10 in
of 18
Table 2.

Table 2. Specifications of the key components.

Table 2. Specifications of the key components.
Component Supplier Type Specifications
Component Supplier Type Specifications
Power: 50 mW,
Light source Minghui Optics M650D50-20100 Power: 50 mW,
Wavelength: 650 nm,
Light source Minghui Optics M650D50-20100 Wavelength: 650 nm,
Spot diameter: 15 mm.
Spot diameter: 15 mm.
Material: K9 glass,
Right-angle prism Daheng Optics GCL-030107A Material: K9≤glass,
Flatness: 0.06 µm,
Right-angle prism Daheng Optics GCL-030107A Dimension
Flatness: (mm): 30 × 30 × 30.
≤0.06 µm,
Sanfeng DimensionMaterial:
(mm): 30 × 30 × 30.
K9 glass,
Standard Material: K9 glass,
Optical flat 120 × 30 × 25 Flatness: ≤0.05 µm,
Sanfeng Standard Measuring
Measuring
Optical flat Implements
120 × 30 × 25 Flatness:
Dimension µm,× 30 × 25.
≤0.05 120
(mm):
Implements
Dimension (mm): 120 × 30 × 25.
Travel range: 50 mm,
Linear stage Zolix KSA050-12-Z Travel range:
Positioning 50 mm,
accuracy: ≤±3 µm,
Linear stage Zolix KSA050-12-Z Positioning accuracy:≤≤±3
Straightness: µm,
10 µm.
Straightness: ≤10 µm.
Measuring range: 600 µm,
Confocal sensor Precitec CHRocodile SE Measuring range:
Linearity error:600 µm,
<0.2 µm,
Confocal sensor Precitec CHRocodile SE Linearity Resolution:
error: <0.2 3 nm.
µm,
Resolution: 3 nm.
Magnification: 0.5~0.8,
Focal
Magnification:length: 50 mm,
0.5~0.8,
Lens Moritex ML-MC50HR
Maximum compatible target:
Lens Moritex ML-MC50HR Focal length: 50 mm,
2/3”.
Maximum compatible target: 2/3″.
Target size: 2/3”,
Camera Basler acA2000-165um
Target size: 2/3″,
Frame rate: 165 fps,
Camera Basler acA2000-165um Frame rate:
Resolution: 165
2048 × fps,
1088. (2 MP)
Resolution: 2048 × 1088. (2 MP)

Figure 11. The experimental platform.

Figure 11. The experimental platform.
5.1. System Calibration
The optical flat was mounted on the linear stage as a geometric reference and formed
Before with
an air wedge verifying the straightness
the stationary errorprism.
right-angle measurement
The two capability of the
mirrors were usedfilm interfer-
to adjust
ometer, the linear coefficient c was determined. From Equation (15), c was mainly
the incident light. The linear stage was driven by an MC600 controller (Zolix Instruments affected
byLtd.,
Co., the wavelength λ of the
Beijing, China). incident
During beam,
the stage the refractive
movement, indexes n1 images
interferogram and n2, were
and the incident
captured
at angle θ1. It was
51 locations difficult
with to obtain
an increment of 1the four
mm. parameters
The accurately.
confocal sensor In this
was used as study, therefore,
a reference for
c was determined experimentally. By precisely adjusting the distance between the two
optical surfaces with known positions, the coefficient c could be obtained.
The measurements were repeated five times. As shown in Figure 11, the confocal
sensor was used to record the change in distance Δh. Assuming that the measurement lo-
Photonics 2021, 8, 149 11 of 18

calibration and comparison. Since only about 5% of the measurement range was used, it
could be considered that the nonlinearity error in this range was insignificant.

5.1. System Calibration

Before verifying the straightness error measurement capability of the film interferome-
ter, the linear coefficient c was determined. From Equation (15), c was mainly affected by
the wavelength λ of the incident beam, the refractive indexes n1 and n2 , and the incident
angle θ1 . It was difficult to obtain the four parameters accurately. In this study, therefore,
Photonics 2021, 8, x FOR PEER REVIEW
c was determined experimentally. By precisely adjusting the distance between the11two of 18
optical surfaces with known positions, the coefficient c could be obtained.
The measurements were repeated five times. As shown in Figure 11, the confocal
sensor was used
introduced to record
in Section the change
4.2. Then in distance
the relationship ∆h. Assuming
between that the measurement
distance variation and phase shift
lo-cation of the as
was obtained, film interferometer
shown in Figure 12. wasThe
in the center
phase shiftofwas
the approximately
image (1024, 544), thewith
linear phasethe
shift of the interferogram ∆ϕ was calculated by the developed image processing
distance variation, which was consistent with the theoretical analysis (in Section 2) and algorithm,
assimulation
introduced (ininSection
Section3).4.2.
TheThen the relationship
linearity coefficient between distance by
c was calculated variation and the
averaging phase
five
shift was obtained, as shown in Figure 12. The phase shift was approximately
slope values, as shown in Table 3. The standard deviation of c was about 0.1% of the linear withav-
the distance variation, which was consistent with the theoretical analysis (in Section 2)
erage.
and simulation
The c value (incalculated
Section 3).from
The Table
linearity
3 wascoefficient
slightly cdifferent
was calculated
from thebytheoretical
averagingvalue
the
five slope values, as shown in Table 3. The standard deviation of c was about
from Table 1. The main root causes of this deviation included the deviation of the actual 0.1% of the
average.
angle θ1 against the ideal setting and the imperfection of the optical components.

Figure12.
Figure 12.Relationship
Relationshipbetween
betweendistance
distancevariation
variationand
andphase
phaseshift.
shift.

Table 3. Calibration results of the linear coefficient c.

Table 3. Calibration results of the linear coefficient c.
Trial 1 2 3 4 5 Average Standard Deviation
Standard
Slope
Trial (rad/μm)
1 10.461
2 10.456 310.464 10.459
4 10.426
5 10.453
Average 0.013
Deviation
Slope
After the calibration,
10.461 10.456the relationship
10.464 between10.426
10.459 Δφ and Δh10.453
could be obtained
0.013 accord-
(rad/µm)
ing to Equation (16):

The c value calculated from Table∆φ = 10.453

3 was × ∆h.
slightly (20)
different from the theoretical value
from Table 1. The main root causes of this deviation included the deviation of the actual
angle θ1 against the
5.2. Measurement ideal setting and the imperfection of the optical components.
Performance
After the calibration, the relationship between ∆ϕ and ∆h could be obtained according
When the linear relationship was determined, the variation of the distance between
to Equation (16):
the two optical surfaces was calculated by analyzing the phase shift of each interferogram.
∆ϕ = 10.453 × ∆h. (20)
The confocal sensor was applied as a reference sensor. The comparison of the measure-
ment
5.2. results was
Measurement shown in Figure 13a. During the assembly process, it was difficult to
Performance
ensure that the optical flat was strictly parallel to the ideal moving axis. This parallelism
When the linear relationship was determined, the variation of the distance between
error resulted in a linear increase (or decrease) in the thickness of the air wedge. This
the two optical surfaces was calculated by analyzing the phase shift of each interferogram.
thickness variation was superimposed with the straightness error, as shown in Figure 13a,
The confocal sensor was applied as a reference sensor. The comparison of the measurement
and could be removed by the linear fitting process, as shown in Figure 13b. The data of
five repeats showed a strong correlation. As shown in Figure 13c, the measurement devi-
ation between the film interferometer and the reference confocal sensor was within ±0.22
µm. In the travel range of 50 mm, the standard deviation was within 25 nm.
 Photonics 2021, 8, 149 12 of 18

results was shown in Figure 13a. During the assembly process, it was difficult to ensure
that the optical flat was strictly parallel to the ideal moving axis. This parallelism error
resulted in a linear increase (or decrease) in the thickness of the air wedge. This thickness
variation was superimposed with the straightness error, as shown in Figure 13a, and could
be removed by the linear fitting process, as shown in Figure 13b. The data of five repeats
showed a strong correlation. As shown in Figure 13c, the measurement deviation between
Photonics 2021, 8, x FOR PEER REVIEW the film interferometer and the reference confocal sensor was within ±0.22 µm. 12 ofIn
18 the
travel range of 50 mm, the standard deviation was within 25 nm.

(a)

(b)

(c)
Figure 13. Experimental
Figure results:
13. Experimental (a) distance
results: variation;
(a) distance (b) straightness
variation; error; (c)error;
(b) straightness measurement de-
(c) measurement
viation.
deviation.

5.3. Analysis of the Source of Measurement Deviation

In this study, there were three main error sources: measurement equipment, meas-
urement process, and measurement environment. In this section, the error source analysis
and measurement accuracy improvement were discussed.
Photonics 2021, 8, 149 13 of 18

5.3. Analysis of the Source of Measurement Deviation

In this study, there were three main error sources: measurement equipment, measure-
ment process, and measurement environment. In this section, the error source analysis and
measurement accuracy improvement were discussed.
(1) Measurement equipment

Photonics 2021, 8, x FOR PEER REVIEW

As shown in Table 2, the flatness of the right-angle prism and the optical flat were 0.06
13 of 18
µm and 0.05 µm, respectively. Hence, the systematic error caused by the imperfection of
the two optical elements might reach up to a submicron level theoretically. Secondly, the
confocal sensor used as the reference had its own measurement errors, such as nonlinearity
the confocal sensor used as the reference had its own measurement errors, such as non-
errors.
linearity errors.
(2) Measurement process
(2) Measurement process
In the measurement process, the misalignment of the measurement locations of the
In the measurement process, the misalignment of the measurement locations of the
film interferometer and the reference confocal sensor was very critical. Figure 14 shows the
film interferometer and the reference confocal sensor was very critical. Figure 14 shows
impact of the misalignment.
the impact of the misalignment.

(a)

(b)
Figure 14. The
Figure 14. relationship between
The relationship betweenthe
thehorizontal
horizontalalignment
alignmentofofthe
the measurement
measurement locations
locations and
and thethe measurement
measurement devia-
deviation:
tion: (a) the measurement locations were properly aligned horizontally; (b) the measurement locations were not properly
(a) the measurement locations were properly aligned horizontally; (b) the measurement locations were not properly aligned.
aligned.
During the movement of the linear stage, the angular motion error was inevitable. As
During
described the movement
in Sections 2 and 3,ofthe
the linear motion
angular stage, the angular
error did notmotion error
affect the was inevitable.
linearity coefficient
As
c, but it varied the distance between the measurement locations of the confocallinearity
described in Sections 2 and 3, the angular motion error did not affect the co-
sensor and
efficient c, but it varied the distance between the measurement
the film interferometer, resulting in a measurement deviation. locations of the confocal
sensorAsand the film
shown interferometer,
in Figure 14a, when resulting in a measurement
the measurement locations deviation.
were well aligned, the
As showndeviation
measurement in Figure due 14a, when
to the the measurement
angular motion error θlocations
x (θx → were
0) could well aligned, the
be expressed as:
measurement deviation due to the angular motion error θx (θx → 0) could be expressed as:
w
∆x = w0 − w = w − w. (21)
∆ =w −w= cos θx − w. (21)
cosθ
∆x 1−
   
coscosθ
θx 11 θθx
lim lim =∆ w=lim w lim
1− = w lim
= 2 wθlim =0 (22)
θx →0 θx→
θ θx →0 → θxθcoscosθ θx 2 x→ cos θx = 0
→ 0 cosθ
(22)
According to
According to Equation
Equation (22),
(22), the
the measurement
measurement deviation ∆xx is
deviation △ is the
the infinitesimal
infinitesimal of
of aa
higher order of angular error θx . It indicated that the measurement deviation caused by the
higher order of angular error θx. It indicated that the measurement deviation caused by
angular error was insignificant when the measurement locations were properly aligned.
the angular error was insignificant when the measurement locations were properly
aligned.
As shown in Figure 14b, when the measurement locations of the confocal sensor and
the film interferometer were not aligned horizontally (the horizontal distance is lx), the
measurement deviation would be different from the above scenario:
 Photonics 2021, 8, x FOR PEER REVIEW 14 of 18
Photonics
Photonics 2021,
2021, 8, 8,
149x FOR PEER REVIEW 1414
ofof
1818

As shown in Equation (24), the measurement deviation △x is the infinitesimal of the

same As shown
As order
shown inFigure
Equation
ofinangular 14b, (24),
error the
θx. It
when measurement
means
the deviation
that the impact
measurement of the
locations △ is the
ofx angular
the infinitesimal
errorsensor
confocal of
was signifi-
and the
same
cant. order of
Therefore, angular
in order error
to θ x. It means
accurately that
evaluate thetheimpact of
straightness
the film interferometer were not aligned horizontally (the horizontal distance is lx ), the the angular
error error
measurementwas signifi-
capa-
cant.
bilityTherefore,
measurement of the film in order to accurately
interferometer,
deviation would was evaluate
be itdifferentnecessary the
from the straightness
to above
ensurescenario:
that the error measurement
measurement capa-
locations
bility of the film interferometer, it was necessary to ensure
of the film interferometer and the confocal sensor were aligned horizontally and verti- that the measurement locations
w
of the film interferometer
cally. ∆x = and w0 −the w= confocal sensor + lx ×were tan(θaligned
x ) − w. horizontally and verti- (23)
cally.Figures 15 and 16 show the method cos(θx )
to ensure the proper alignment. I1, I2, …, I50 were
the 50 Figures 15 and 16 captured
interferograms show
∆x theby method to ensure
the camera, θx andthe D1,proper
 alignment. I1, I2, …, I50 were
D2,θ…,
tan x D50 were the distance varia-
the
tion50measured
interferograms
by the limcaptured = 0 sensor.
+
bylxthelim camera, and = lxDlim 1, D2, …, D= lx
50 were the distance varia- (24)
θx →confocal
0 θx θx → 0 θ x θx →0 θx
tion measured by the confocal sensor.
When the image processing algorithm was executed, every image (the fifth in Figure
As shown
When theinimage
Equation (24), the measurement
processing deviation ∆x isimage the infinitesimal inof the
9) was equally divided into four algorithm
quadrants—Q was 1executed,
, Q2, Q3, Qevery4—and their (the fifth
center points—OFigure 1,
same
9) order of angular error θx . It means that the impact of the angular error was significant.
O2was
, O3,equally
O4—were divided into four
determined, quadrants—Q
respectively. 1, Q2these
Taking , Q3, Qfour
4—and centerstheirascenter points—O1,
the measurement
Therefore,
O in order determined,
2, O3, O4—were
to accuratelyrespectively.
evaluate the Taking straightness
theseerror
four measurement
centers as the capability
measurement of
locations of the film interferometer, four measurement deviation results—E 1, E2, E3, E4—
the film
locations interferometer, it
of the filmComparing was necessary
interferometer, to ensure
fourvalues,
measurement that the measurement locations of the
could be obtained. the four assumingdeviation
E1 was the results—E
minimum 1, E 2, E3, E
value, 4—
then
film
could interferometer
be obtained. and the confocal
Comparing the sensor
four wereassuming
values, aligned horizontally
E was the and vertically.
minimum value, then
O1 was the closest to the optimal measurement location among the four center points. In 1

O Figures
1 waswords,
15 and to
the closest 16theshow the method
optimal measurementto ensure the proper alignment. I1 , I2 ,points.
. . . , I50In
other the optimal measurement locationlocation
was in Q among the four center
1. Subsequently, Q1 was further
were
other thewords,
50 interferograms
the optimal captured
measurement by thelocation
camera,was andin D1Q, 1D 2 , . . . , D50 wereQthe
. Subsequently, 1 was
distance
further
divided into four quadrants, and the above process was repeated. After several iterations,
variation measured by the confocal sensor.
divided into four quadrants, and the above process
the measurement location could be sufficiently close to the optimal location. was repeated. After several iterations,
the measurement location could be sufficiently close to the optimal location.

Figure 15. The measurement locations alignment method based on successive approximation.
Figure 15. The measurement locations alignment method based on successive approximation.
Figure 15. The measurement locations alignment method based on successive approximation.

Figure 16. The process of measurement locations alignment.

Figure 16. The process of measurement locations alignment.
Figure 16. The process of measurement locations alignment.
Photonics 2021, 8, 149 15 of 18

When the image processing algorithm was executed, every image (the fifth in Figure 9)
was equally divided into four quadrants—Q1 , Q2 , Q3 , Q4 —and their center points—O1 ,
Photonics 2021, 8, x FOR PEER REVIEW 15 of 18
O2 , O3 , O4 —were determined, respectively. Taking these four centers as the measurement
locations of the film interferometer, four measurement deviation results—E1 , E2 , E3 , E4 —
could be obtained. Comparing the four values, assuming E1 was the minimum value, then
O1 was the closest
Therefore, into thestudy,
this optimalthemeasurement location
alignment process didamong the four
not require anycenter points.
physical In
adjust-
other
ment.words, the optimal
The alignment measurement
process was actuallylocation was in Qof1 .the
the selection Subsequently, Q1 was further
optimal measurement point
divided into four
in the image. quadrants,
Compared andthe
with thesingle
above point
process was repeated.
sensing methods,After
suchseveral
as LVDTiterations,
(Linear
the measurement
Variable location
Differential could be sufficiently
Transformer), the assemblycloseand
to the optimal location.
adjustment processes were much
Therefore, in this study, the alignment process did not require any physical adjustment.
simpler.
The alignment process was
After identifying actuallymeasurement
the optimal the selection of the optimal
location, measurement
the same experimentspoint in the
in Section
image. Compared with the single point sensing methods, such as LVDT
5.2 were performed again. The experimental data are shown in Figure 17. The measure- (Linear Variable
Differential
ment deviationTransformer),
between the theassembly and adjustment
film interferometer processes
and the were
reference much simpler.
confocal sensor was
Afterto
reduced identifying
±0.1 µm, the
whichoptimal measurement
indicated location,
a significant the same experiments
improvement, as compared in Section
to ±0.225.2
µm
were performed again. The experimental data are shown in Figure 17.
shown in Figure 13c. Five repeats were performed at each location. The standard devia- The measurement
deviation
tion was between
within 25the nm.film interferometer and the reference confocal sensor was reduced
to ±0.1 µm, which indicated
Considering the imperfect a significant
flatness ofimprovement, as compared
the optical flat to ±0.22
and the inherent µm shown
measurement
inerror
Figure 13c. Five repeats were performed at each location. The standard
from the confocal sensor, ±0.1 µm was a conservative accuracy statement of the deviation wasde-
within 25 nm.
velopment system.

Figure17.
Figure 17.Measurement
Measurementdeviation
deviationcalculated
calculatedbased
basedon
onthe
thenew
newlinear
linearcoefficient
coefficientc.c.

It’s worth highlighting

Considering the imperfectthat this process
flatness of selecting
of the optical the the
flat and optimal measurement
inherent measurement loca-
tion was
error fromcritical for not only
the confocal the ±
sensor, calibration
0.1 µm was process but also theaccuracy
a conservative actual measurement
statement ofwhen
the
setting up the
development film interferometer into a topography measurement shown in Figure 1(b).
system.
In actual topography
It’s worth measurement,
highlighting the main
that this process probe measuring
of selecting the optimal themeasurement
height wouldlocation
take the
was critical
place of thefor not only
reference the calibration
confocal sensor shownprocess but also
in Figure 11.the
Theactual measurement when
film interferometer should
setting
be ableup tothe film interferometer
quantify and compensate intofor
a topography
the motion measurement
error that occurredshownininthe
Figure
axial1b. In
direc-
actual
tion oftopography
the main probe.measurement, the main probe measuring the height would take the
place of the reference
(3) Measurement environmentconfocal sensor shown in Figure 11. The film interferometer should
be able to quantify and compensate for the motion error that occurred in the axial direction
The temperature was a concern for most interferometric measurement techniques. A
of the main probe.
temperature drift experiment was performed to verify the environment robustness of the
(3)
film Measurement
interferometer. environment
The film interferometer was placed in a common air-conditioned en-
vironment for one
The temperature was hour, and the camera
a concern for was
mostset to take an interferogram
interferometric measurement every 30 s. Based
techniques.
A temperature drift experiment was performed to verify the environment robustnessdrift
on the phase analysis algorithm described in Section 4.2, the measurement reading of
was
the obtained.
film As shown
interferometer. TheinfilmFigure 18, the reading
interferometer drift of
was placed in the film interferometer
a common air-conditionedwas
within 12 nmfor
environment in one hour
hour,when
and the thecamera
temperature
was set varied between
to take 18 °C and 22every
an interferogram °C. 30 s.
Based on the phase analysis algorithm described in Section 4.2, the measurement reading
Photonics 2021, 8, 149 16 of 18

drift
Photonics 2021, 8, x FOR PEER REVIEW was obtained. As shown in Figure 18, the reading drift of the film interferometer16
was
of 18
within 12 nm in one hour when the temperature varied between 18 ◦ C and 22 ◦ C.

Figure 18. Reading drift of the film interferometer.

Figure 18. Reading drift of the film interferometer.

6.6.Conclusions
Conclusions
InInthis
thispaper,
paper,a afilm
filminterferometer
interferometerfor formeasuring
measuringthe thestraightness
straightnesserrorerrorofofa aprecision
precision
motion
motionsystem
systemwas wasproposed.
proposed.The Theprinciple
principleofofthethefilm
filminterference
interferencemodulemoduleand andphase
phase
calculation
calculationprocess
processwerewerediscussed.
discussed.InInthe thepresented
presentedsystem,
system,an anoptical
opticalflatflatand
anda aright-
right-
angle
angleprism
prismwere
wereusedusedtotogenerate
generatethe theinterferogram.
interferogram.An Animage
imageprocessing
processingmethodmethodwas was
applied to calculate the phase shift induced by the straightness error.
applied to calculate the phase shift induced by the straightness error. The components The components used
inused
this in
system were widely
this system available
were widely and affordable
available in the market.
and affordable In addition,
in the market. compared
In addition, com-
with
paredtraditional interferometric
with traditional systems, systems,
interferometric the presented system was
the presented also featured
system was also by its easyby
featured
alignment process and
its easy alignment goodand
process environmental robustness.
good environmental robustness.
Experimental
Experimentaltests testswere
were performed
performed to to verify
verify the
thestraightness
straightnesserror errormeasurement
measurement ca-
capability of the developed film interferometer. The experimental results
pability of the developed film interferometer. The experimental results showed that the showed that the
measurement
measurementdeviation
deviationbetween
betweenthe theproposed
proposedsystem
systemand andthethereference
referencesensor
sensorwas wasbetter
better
than ±
than ±0.1 µm, and the repeatability, represented by the standard deviation, was within2020
0.1 µm, and the repeatability, represented by the standard deviation, was within
nmnminina atravel
travelrange
rangeofof5050mm. mm.
This
This system is potentiallyable
system is potentially abletotomeasure
measurethe thestraightness
straightnessininbothbothlinear
linearandandangular
angular
values online. At present, the main focus is on the analysis of its linear
values online. At present, the main focus is on the analysis of its linear value measurement value measure-
ment capability.
capability. The new
The new capability
capability of measuring
of measuring the angular
the angular motion motion
errorerror
is alsoisunder
also under
devel-
development.
opment.
This
Thisproposed
proposedfilm filminterferometer
interferometer was successfully implemented
was successfully implementedininaa3D 3Dsurface
surfaceto-
topography
pography measurement system developed by the authors’ team, as shown in Figure19.
measurement system developed by the authors’ team, as shown in Figure 19.
In that system, the film interferometer was used to compensate for the motion error. With
In that system, the film interferometer was used to compensate for the motion error. With
effective error compensation, the 3D surface topography measurement system was able
effective error compensation, the 3D surface topography measurement system was able
to achieve repeatability and reproducibility within 0.1 µm. The objective of publishing
to achieve repeatability and reproducibility within 0.1 µm. The objective of publishing
this paper was to provide more details of the film interferometer, which was applied as a
this paper was to provide more details of the film interferometer, which was applied as a
subsystem in the work published earlier.
subsystem in the work published earlier.
Photonics 2021, 8, 149 17 of 18
Photonics 2021, 8, x FOR PEER REVIEW 17 of 18

(a) (b)
Figure 19.
Figure 19. System
System configuration:
configuration: (a)
(a) 3D
3D design;
design; (b)
(b) actual
actual setup.
setup.

Author Contributions: Conceptualization, F.C.; methodology, F.C.; software, H.S.; validation,

R.Y.,
AuthorF.C., C.C., and Q.Y.;
Contributions: formal analysis, H.S.,
Conceptualization, F.C.;R.Y., and F.C.; investigation,
methodology, F.C.; software,H.S. and
H.S.; R.Y.; re- R.Y.,
validation,
sources, R.Y. and F.C.; data curation, H.S. and R.Y.; writing—original draft preparation,
F.C., C.C. and Q.Y.; formal analysis, H.S., R.Y. and F.C.; investigation, H.S. and R.Y.; resources, H.S. and
R.Y.
R.Y.; writing—review & editing, H.S., R.Y., F.C., C.C., and Q.Y.; visualization, H.S. and
and F.C.; data curation, H.S. and R.Y.; writing—original draft preparation, H.S. and R.Y.; writing—F.C.; su-
pervision, R.Y. andH.S.,
review & editing, F.C.;R.Y.,
project
F.C.,administration,
C.C. and Q.Y.; R.Y. and F.C.; funding
visualization, H.S. andacquisition, H.S., R.Y.,
F.C.; supervision, R.Y.and
and
F.C. All authors have read and agreed to the published version of the manuscript.
F.C.; project administration, R.Y. and F.C.; funding acquisition, H.S., R.Y. and F.C. All authors have
read and agreed
Funding: to the published
This research version
was funded by theofFundamental
the manuscript.
Research Funds for the Central Universi-
ties,
Funding: This research was funded by the Fundamental Researchof
China (ZQN-703); the National Natural Science Foundation China
Funds for(Grant No. 51605171);
the Central Universities,
the National Natural Science Foundation of China (Grant No. 52075190);
China (ZQN-703); the National Natural Science Foundation of China (Grant No.and the Subsidized Pro-
51605171); the
ject for Postgraduates’ Innovative Fund in Scientific Research of the Huaqiao University (Grant
National Natural Science Foundation of China (Grant No. 52075190); and the Subsidized Project
No. 18013080058). Innovative Fund in Scientific Research of the Huaqiao University (Grant No.
for Postgraduates’
18013080058).Review Board Statement: Not applicable.
Institutional
Institutional
Informed Review
Consent Statement: InformedNot
Board Statement: applicable.
consent was obtained from all subjects involved in the
study.
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.
Data Availability Statement: Data available in a publicly accessible repository.
Data Availability Statement: Data available in a publicly accessible repository.
Acknowledgments: The authors would like to thank LetPub (www.letpub.com) for its linguistic
Acknowledgments: The authors would like to thank LetPub (www.letpub.com, accessed on 19 April
assistance during the preparation of this manuscript and all reviewers for their helpful comments
2021) for its linguistic assistance during the preparation of this manuscript and all reviewers for their
and suggestions.
helpful comments and suggestions.
Conflicts of Interest: The authors declare no conflicts of interest.
Conflicts of Interest: The authors declare no conflict of interest.

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