Optical Review (2021) 28:425–433
https://doi.org/10.1007/s10043-021-00675-3

REGULAR PAPER

Birefringence compensation for single‑shot 3D profilometry using
a full‑Stokes imaging polarimeterYuuki Maeda1· Shuhei Shibata1 · Nathan Hagen1 · Yukitoshi Otani1

Received: 17 April 2021 / Accepted: 22 June 2021 / Published online: 28 June 2021
© The Optical Society of Japan 2021

Abstract
We demonstrate a single-shot uniaxial 3D profilometry system that illuminates a sample with a structured polarization
pattern and measures the reflected full-Stokes polarimetric image. Some non-contact 3D measurements require evaluating
samples located in a protected environment. For example, this may be a sample under the lid of an acrylic Petri dish, a sample that has a protective film overlay, or products housed inside a plastic container. To deal with these cases, we adapt ourmeasurement system to capture the complete polarization state using a single-shot full-Stokes imaging system consisting of
two polarization cameras. This system cannot only measure the dynamic 3D shape at video rate, but can also compensate
for any birefringent layers between the illumination arm and the sample under measurement. We present the measurement
principles of the system and show the results of a real-time 3D profilometry experiment in which a sample was measured
under the lid of an acrylic Petri dish.
Keywords Profilometry · Polarization camera · Fringe projection · Single-shot measurement · Polarization pattern

1 Introduction
Optical methods for three-dimensional surface profile measurement are used in a wide range of fields such as manufacturing inspection, medical inspection, and robot vision,
because they allow non-contact and non-destructive measurement. Stereo optical techniques such as the light-section
method [1], the Moiré method [2], and the fringe projection method [3–5] work well for surfaces that do not haveextreme gradients. Samples with deep holes or large stepheights cause difficulties for these methods because someregions are occluded from the projector or from the imager.
In these cases, uniaxial profilometry methods such as theTime-of-Flight (TOF) method, or the focus method, aremore practical. The TOF method projects pulse-modulated
light from a light source onto an object, and measures the
distance from the time it takes for pulses to return froman object. This method is well suited to measuring large* Yuuki Maeda
dt187180@cc.utsunomiya-u.ac.jp
1

Center for Optical Research and Education, Utsunomiya
University, 7‑1‑2 Yoto, Utsunomiya, Tochigi 321‑8585,
Japan

objects, but has difficulty resolving small feature sizes, suchas in the micrometer or millimeter range.
The focus method for uniaxial profilometry obtains thesurface height from changes in the contrast (due to blurring)of a projected pattern. The three-dimensional informationof the sample can be obtained by measuring the relationship between height and contrast in advance. Neil et al. usedphase shifting with fringe projection methods to perform
optical sectioning and determined the change in fringe
contrast with focus [6]. Ishihara et al. proposed using thecontrast dependence on focus to perform 3D profilometry[7–9]. However, the fringe pattern method cannot measure
the surface profile in a single snapshot, because it requires
multiple shifts of the illumination pattern to obtain the contrast. To measure in a snapshot, we have previously proposeda uniaxial three-dimensional profilometry system which illuminates a sample with a spatially varying polarization pattern and measures the spatial distribution of contrast with apolarization camera [10]. However, if there is a birefringent
element within the illumination path, the resulting changes
in the polarization pattern cause errors in the estimated surface height. For example, this may be a sample under the lidof an acrylic Petri dish or a sample that has a protective film
overlay. To deal with these cases, we adapt our measurement
system to capture not just the linear Stokes vector elements

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Optical Review (2021) 28:425–433

but the complete polarization state using a full-Stokes imaging system (FSIS) consisting of two polarization cameras[[11, 12]. Since]T the FSIS can measure the full-Stokes vector
s0 , s1 , s2 , s3 in a snapshot, this profilometry system can not
only measure dynamic profiles but can also compensate forany birefringent layers between the illumination arm and thesample under measurement.

2 Experimental setup and measurement
principles
The system measures the degree of polarization (DOP) of
the light reflected from a sample using a spatially varying
polarization pattern and a FSIS. Figure 1 shows the experimental setup of a uniaxial 3D profilometry system using the
FSIS and the conventional setup [10].
The FSIS consists of two polarization cameras and aquarter-wave plate (QWP), where the QWP is oriented at
an azimuth angle of 45° (see Fig. 1a) [11]. Each polarization camera (P1, P2) in the imaging system is composedof a micropolarizer array and detector array, where theFig. 1  a Experimental setup
using the full-Stokes imaging
system. b Conventional setup(a)micropolarizers are oriented at the four azimuth angles of
0°, 45°, 90°, and 135°.
The projection system produces the spatially varying polarization pattern using a QWP and a spatial light
modulator (SLM) [10]. Starting from the light source, laser
light is scattered by a rotating diffuser and collimated bylens L1. The rotating diffuser reduces speckle noise in
the detected image, while L1 produces a collimated beam
diameter matched to the SLM and objective lens L3. The
collimated light is linearly polarized by linear polarizer LP
and directed towards the SLM by the non-polarizing beam
splitter (NPBS). The linear polarization pattern is made by
the SLM and the QWP.
We can characterize the linear polarization pattern quantitatively using Mueller calculus. Passing the input Stokes
vector sin from the light source (which we assumed to be
unpolarized) through each of the optical elements in turngives the output Stokes vector sout projected onto the sample
given by

FSIS

(b)

P1
QWP

P2

P1

NPBS

L4

L4

SLM
QWP

NPBS

NPBS

LP
L1

NPBS

L2

L2
SampleL3

L3

Diffuser
SampleLid

Light Source

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Acrylic petri dish

Optical Review (2021) 28:425–433

427

�
�
�
�
� �
� �
sout (x, y) = QWP 135◦ ⋅ SLM 𝜙(x, y), 0◦ ⋅ QWP 45◦ ⋅ LP 0◦ ⋅ sin

1⎤⎡⎢ cos{𝜙(x, y)} ⎥sout = ⎢⎥.
⎢ sin{𝜙(x, y)} ⎥⎣⎦0⎡1 0 0 0⎤⎢⎥⎢0 0 0 1⎥=⎢⎥⎢0 0 1 0⎥⎢ 0 −1 0 0 ⎥⎣⎦⎤⎡ rs + rp rs − rp
0
0
⎥
⎢
0
0
⎥
⎢ rs − rp rs + rp
√
√
⎥
⎢
r
cos
y)}
2
r
r
r
sin
y)}
0
0
2
{𝜙(x,
{𝜙(x,
s p
s p
⎥
⎢
√
√
⎢ 0
0
−2 rs rp sin {𝜙(x, y)} 2 rs rp cos {𝜙(x, y)} ⎥⎦⎣⎡ 1 0 0 0 ⎤ ⎡ 1 1 0 0 ⎤⎡ 1 ⎤
⎢
⎥ ⎢
⎥⎢ ⎥
⎢ 0 0 0 −1 ⎥ 1 ⎢ 1 1 0 0 ⎥⎢ 0 ⎥
⎢
⎥ ⎢
⎥⎢ ⎥
⎢ 0 0 1 0 ⎥ 2 ⎢ 0 0 0 0 ⎥⎢ 0 ⎥⎢ 0 1 0 0 ⎥ ⎢ 0 0 0 0 ⎥⎢ 0 ⎥
⎣
⎦ ⎣
⎦⎣ ⎦

⎤
⎡
rs + rp⎥⎢ √1 ⎢ 2 rs rp cos {𝜙(x, y)} ⎥,= ⎢ √2 ⎢ 2 rs rp sin {𝜙(x, y)} ⎥⎥⎢⎥rs − rp⎣⎦(1)

where rs , rp , and 𝜙 represent the s- and p-reflectance fromthe SLM, and the phase applied to the SLM and (x, y) represents the two-dimensional coordinates of the image sensor
surface of the polarization camera. If the angle of incidenceof the light on the liquid crystal panel is small, the s- andp-reflectance from the SLM are almost equal. If we substitute rs = rp = 1 in Eq. (1), sout becomes(a)First, we consider the case where there is no phase element between the illumination path and the sample. In thiscase, the Stokes vector simg measured by the detector array
will be0
0 ⎤⎡ s0 ⎤ ⎡ 1 0⎢ s1 ⎥ ⎢ 0 A(x, y) 00 ⎥simg = ⎢ ⎥ = ⎢⎥⎢ s2 ⎥ ⎢ 0 0 A(x, y) 0 ⎥⎣ s3 ⎦ ⎣ 0 0
0 A(x, y) ⎦1⎡⎤
⎢ A(x, y) cos {𝜙(x, y)} ⎥sout = ⎢⎥,⎢ A(x, y) sin {𝜙(x, y)} ⎥
⎦
⎣
0

(3)

where A(x, y) represents the contrast.
The contrast is estimated using the phase-shift method.
From Eq. (3), we see thatangle of linear polar( the azimuthal)
ization, given by tan−1 s2 ∕s1 , is equal to the phase 𝜙(x, y)of the SLM. The spatially varying angle of polarization iscontrolled simply by adjusting the phase 𝜙(x, y) at each column across the SLM. The light intensity I(x, y, 𝜓) detectedby the polarization camera is given by

(b)
1.0

0.04

Error (mm)

DOP

(2)

0.5

0.02
0.00
-0.02

0.0

-2.0

-1.0

0.0

1.0

Height- Z (mm)

2.0

-0.04

0

10

20

30

Measurement number

Fig. 2  a The DOP as a function of the sample height. b The error of the measured height at height 0.5 mm

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Optical Review (2021) 28:425–433

where P(𝜓) is the Mueller matrix of a micropolarizer oriented at angle 𝜓 ∈ {0◦ , 45◦ , 90◦ , 135◦ }, and d is the detection vector. Parameters A(x, y) and 𝜙(x, y) are calculated as
√
√(
)2 (
)2
s21 + s222
I0 − I90 + I45 − I135
(5)=A(x, y) =I0 + I45 + I90 + I135
s0

I(x, y, 𝜓) = dT P(𝜓) ⋅ simg
�1�
= 10002
1
cos(2𝜓)sin(2𝜓)0⎤⎡⎢ cos(2𝜓)cos2 (2𝜓)cos(2𝜓) sin(2𝜓) 0 ⎥⎢ sin(2𝜓) cos(2𝜓)sin(2𝜓)
0 ⎥⎥sin2 (2𝜓)
⎢
⎣
0
0
0
0⎦
1
⎡
⎤
⎢ A(x, y)cos{𝜙(x, y)} ⎥⎢ A(x, y)sin{𝜙(x, y)} ⎥
⎢
⎥
⎣
⎦
0

1
= {1 + A(x, y) cos (𝜙(x, y) − 2𝜓)},2

𝜙(x, y) = tan(

I45 − I135
I0 − I90

)

−1

= tan

(

)
s2
,
s1

(6)

I0 = I(x, y, 0◦ ), I45 = I(x, y, 45◦ ), I90 = I(x, y, 90◦ ),I135 = I(x, y, 135◦ ).
From Eq. (5), for a polarization invariant sample (i.e.,
diagonal Mueller matrix), the contrast of the phase-shift

(4)

(b)

(a)
0.3
0.2

B

A

0.1
0.2

DOP / DOLP

1.0

X (mm)

0.6

0.2

A

B

0.4

0

0.4

0.2

0.2

X (mm)

0

0.4

DOP

0.5

DOLP

0

(d)

0.8

0.1

0

(c)

1

0.3

Y(mm)

Y(mm)

Fig. 3  a In-plane DOP distribution measured by this system.
b In-plane DOLP distribution
measured by the conventional
system. c Cross-section of the
DOP and DOLP. d 3D profile
measured from DOP. e 3D profile measured from DOLP

−1

0.2

X (mm)

0.4(e)

Z (mm)Z (mm)

2
(mm)

1
2

0.5

X (mm)

0.4

Y (mm)2

0.5

X (mm)

0.4
Y (mm)

0-1

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Optical Review (2021) 28:425–433

429

method is equal to degree of linear polarization (DOLP) of
the linear polarization illumination pattern. Therefore, we
can measure the contrast in a single shot by estimating the
DOLP at each pixel in the polarization camera, using thefour polarization-projection images (I0 , I45 , I90 , I135 ). For apolarization invariant sample, the contrast A(x, y) is a fixed
function of the distance of the sample surface from focus[10]. The three-dimensional coordinates of the sample are
given by

Z(X, Y) = f (A(X, Y)) = f (A(x, y)),(7)

where (X, Y, Z) represents the three-dimensional coordinates at the sample surface. If the optical system is telecentric, then X ∝ x , Y ∝ y for all Z.
Next we consider the case where the sample lies underneath a birefringent phase element (such as the lid of anacrylic Petri dish) that changes the phase of the linear polarization pattern (Fig. 1a). If the sample itself is polarization
invariant, the Stokes vector of the reflected light from thesample will be given by�⎡ s0 ⎤⎢ s� ⎥
�
simg = ⎢ 1� ⎥ = PE(𝜋 − 𝛼, 𝛽) ⋅ PE(𝛼, 𝛽) ⋅ simg ,⎢ s2� ⎥
⎣ s3 ⎦(8)and

3 Experimental results
For the experiments, we chose lens focal lengths for L1, L2,
L3, and L4 of 30, 100, 20, and 200 mm to measure a sample
in the millimeter range. The focal length of L1 is selected sothat the diameter of the collimated light matches the effective area of the SLM. The focal length of L2 matches the
distance between the L2 and the SLM. To collimate the lightfrom the SLM. The focal length of L3 and L4 are determinedfrom the system field of view at the sample, because theratio of the system field of view to the image size of thepolarization camera is equal to the ratio of the focal lengthof L3 to focal length of L4. The optical system focal ratio
is f/10. The light source is a 532 nm Nd–YAG laser. The
SLM is a liquid–crystal-on-silicon type with a 792 × 600
grid of 20 µm pixels (X10468, Hamamatsu Photonics K.K.).
The polarization camera has a 2064 × 1560 array of 2.5 µm
pixels and 30 Hz frame rate. The system field of view is0.51 × 0.39 mm.
We first measure a planar mirror lying underneath the lidof an acrylic Petri dish. Figure 2a shows the DOP measurements obtained while adjusting the height of the sample. The
DOP value (the contrast) is maximum at the best focus position. Since it is not possible to discriminate positive fromnegative distances from the contrast alone, we will use only
one side of the DOP vs. height curve, and measure heightsfrom 0.3 to 1.8 mm. Figure 2b shows the height measure-

0
0
0
⎤
⎡1
⎢ 0 cos2 2𝛼 + sin2 2𝛼 ⋅ cos 𝛽 sin 2𝛼 ⋅ cos 2𝛼(1 − cos 𝛽) − sin 2𝛼 sin 𝛽 ⎥PE(𝛼, 𝛽) = ⎢⎥,22⎢ 0 sin 2𝛼 ⋅ cos 2𝛼(1 − cos 𝛽) sin 2𝛼 + cos 2𝛼 ⋅ cos 𝛽 cos 2𝛼 sin 𝛽 ⎥⎦⎣0
sin 2𝛼 sin 𝛽− cos 2𝛼 sin 𝛽
cos 𝛽
where PE(𝛼, 𝛽) is the Muller matrix of the birefringent
phase element layer, while 𝛼 and 𝛽 are the phase element’s
fast axis and retardance.
The DOLP and the degree of polarization (DOP) of the
Stokes vector s′img are calculated as
√
s�1
+ s�2
1
2
(9)
= A(x, y) ⋅ g(𝛼, 𝛽, 𝛿(x, y)),DOLP =
�
s0

DOP =

√

s�2
+ s�2
+ s�2
1
2
3
s�0

= A(x, y).

(10)Thus, we see that while the DOP is equal to contrast
A(x, y), our previous measurement approach relied on the
DOLP, which contains an error factor when a birefringent
layer is present. In addition, since the FSIS can measure the
DOP in a snapshot, we use this to estimate the contrast, andthus the sample height, in a snapshot.

ment error occurring at a height (distance from best focus) of
0.5 mm and an image size 1032 × 780. The standard deviation of the single frame measurement precision is 0.005 mm,so that we can estimate the measurement resolving power
to be 300.
Figure 3 shows a graph comparing the three-dimensional
shape obtained from DOP and the three-dimensional shapeobtained from DOLP. Figure 3a, b compares the DOP measured by this system to the DOLP measured by our previous measurement approach [10] at a height of 0.5 mm. The
in-plane standard deviation of the DOP-based height measurements in Fig. 3d is 0.026 mm. The DOLP-based heightmeasurements, on the other hand, display large-scale error
as a result of the birefringent phase element.
The previous measurement used a planar mirror sample todemonstrate the differences between using DOLP and DOP
to estimate the sample height when an intermediate layerof birefringent material is present. Next, we measure two
samples under the lid of acrylic Petri dish to demonstratereal-time 3D shape measurements. First, we measured the

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Optical Review (2021) 28:425–433(a) Stepped sample(c) Cross section (A-B)
5
T0

Moving

4

Z

T1

Z (mm)X
Lid↓Mirror-2

3
T2

2
T3

↓Mirror-1

1

Acrylic petri dish

0

0.4

0.2
X (mm)(b) 3D profile
T1: 767ms

T0: 0ms

Z (mm)Z (mm)

3
(mm)

A

A

3.0

3.0

1.5
B
B

Y (mm)X (mm)
0.4

Y (mm)X (mm)
0.4

0.3

0.3

T3: 833ms

T2: 800ms

Z (mm)Z (mm)
A

A

3.0

3.0

B

B

Y (mm)X (mm)
0.4

13

0

0.3

Y (mm)X (mm)

0.4

0.3

Optical Review (2021) 28:425–433◂Fig. 4  Results of the real-time 3D profile measurements of the step

sample under the lid of an acrylic Petri dish: a the stepped sample
under the lid of an acrylic Petri dish. b The 3D profile of the step
sample at time points T0 = 0 ms, T1 = 767 ms, T2 = 800 ms andT3 = 833 ms. c A cross-section of the measured profile at six time
points. The vertical axis gives the surface height for T3, and the othertime points are vertically offset for clarity (see Visualization 1)

moving stepped sample under the lid of an acrylic Petri dish.
The results are shown in Fig. 4. During the experiment, thestepped sample is translated along the x-axis using a linear
stage. The polarization camera captures the data at 30 fps
and the image size at the sample is 0.27 × 0.46 mm. The
thickness of Mirror-1 and Mirror-2 are 1.00 mm. The purpose of this experiment is to measure the moving sample
with large step heights in real time. The height is calculated
from the DOP using the calibrated curve of the reference
plane mirror (Fig. 2a). The average height is 2.24 mm form
x-axis 0–2.0 mm and 1.25 mm form x-axis 3.0–4.6 mm
at time point T2 from Fig. 4c. Second, we measured the3D profile of the moving crosshatch pattern sample under
the lid of an acrylic Petri dish to demonstrate real-time 3D
profilometry, the results of which are shown in Fig. 5. The
purpose of this experiment is to measure the moving sample with deep holes in real time. The height of point P1 is
1.97 mm and the height of point P2 is 1.27 mm at time point
T3 from Fig. 5c. These results show that our setup can bemeasured the shape of the moving sample with large stepheights and/or deep holes.

4 Conclusion
The real-time uniaxial 3D profilometry system we demonstrate here fulfills a need for high precision surface heightmeasurement (0.005 mm precision), video rate data collection (30 Hz frame rate) of object surface profiles underneatha birefringent protective layer such as the lid of an acrylic

431

Petri dish. Our previous method [10] cannot measure thecontrast accurately in this situation, because the contrast
is affected by the change in the linear polarization pattern
induced by the birefringent layer. Here, we use a full-Stokes
imaging system and polarization pattern projection, whichis accurate for a polarization invariant objects as well as
objects having phase difference and objects covered withmaterial with phase difference. This adds another techniqueto the toolbox of engineers designing profilometry systems,
and delivers 3D data at a frame rate limited only by the
speed of the camera. This system gives an advantage formeasuring objects that have deep holes and/or large stepheights covered with material that changes the phase. Therefore, this system can be applied to high-speed measurementof semiconductor bumps and copper pillars. We can measureobjects without accidentally damaging by snapshot.
The system we demonstrate here is suitable for samples
with height variations in the millimeter range, but this canbe adjusted for larger height ranges using larger projection
optics, or for smaller height features using a higher numerical aperture objective lens. For a given SNR, the heightresolving power of the single frame is 300. This system iswell suited to evaluate shapes of a sample composed of a single material covered with a material with a phase differenceor a sample with a phase difference. However, the problemof this system is that it cannot measure a sample composedof materials with different depolarization because the DOP
depends on the depolarization.
Supplementary Information The online version contains supplementary material available at https://d​ oi.o​ rg/1​ 0.1​ 007/s​ 10043-0​ 21-0​ 0675-3.
Acknowledgements This work was supported by JSPS KAKENHI
Grant Number JP 20K04516. This work was partially supported by
JSPS KAKENHI Grant-in-Aid for Scientific Research (S) Number
20H05702.

Declarations
Conflict of interest The authors declare that there are no conflicts ofinterest related to this article.

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Optical Review (2021) 28:425–433(a) Crosshatch pattern sample
Moving(b) Crosshatch pattern
Z
20μm
100μm

X
Lid
Crosshatch pattern
Glass
MirrorAcrylic petri dishMirror Coat(c) 3D profile
T1: 33ms

T0: 0ms

3(mm)

Z (mm)Z (mm)

3.0

3.0

1.5

Y (mm)X (mm)
0.4

Y (mm)X (mm)

0.4

0.3

T3: 100ms

T2: 67ms

0

0.3

P1 ( 0.05 , 0.13 , 1.97)
Z (mm)Z (mm)

P2 (0.30 , 0.13 , 1.27)

3.0

3.0

Y (mm)X (mm)
0.4

0.3

Fig. 5  Results of the real-time 3D profile measurements of the crosshatch pattern sample under the lid of an acrylic Petri dish: a thecrosshatch pattern sample under the lid of an acrylic Petri dish. The
refractive index of the glass is 1.516 and the thickness of the glass

13

Y (mm)X (mm)

0.4

0.3

is 1.03 mm. b The schematic illustration of the crosshatch pattern. c
The 3D profile of the sample at time points T1 = 0 ms to T3 = 100 ms(see Visualization 2)

Optical Review (2021) 28:425–433

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