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Vol. 59, No. 6 / 20 February 2020 / Applied Optics

Single shot 3D profilometry by polarization
pattern projection
Yuuki Maeda, Shuhei Shibata, Nathan Hagen,AND Yukitoshi Otani*

Utsunomiya University, Center for Optical Research and Education, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585, Japan*Corresponding author: otani@cc.utsunomiya-u.ac.jp
Received 11 November 2019; revised 31 December 2019; accepted 3 January 2020; posted 13 January 2020 (Doc. ID 382690);
published 14 February 2020

We demonstrate a uniaxial 3D profilometry system illuminating the sample with a linear polarization pattern
and measuring a polarization camera. The linear polarization pattern is generated by a spatial light modulator
and a quarter-wave plate in the optical system. The system can measure four different fringe patterns with a phasedifference of 90 deg simultaneously in the polarization camera. Therefore, we can measure three-dimensional
shapes in a single shot. We present the measurement principles of the system and show the results of a real-time 3D
profilometry experiment. © 2020 Optical Society of America
https://doi.org/10.1364/AO.382690

1. INTRODUCTION
The fields of manufacturing inspection, medical inspection,
robot vision, etc., have a need for high-precision, high-speed,
and high-resolution measurements of the shapes of objects in
3D. Non-contact profilometry is particularly useful for preserving the surface quality during measurement. Whereas someprofilometers aim to measure surface height features in the
micrometer or nanometer range (surface roughness measurements) [1], the profilometry system that we propose here aims tomeasure samples with surface height variations in the millimeter
range. For this type of measurement, the most common techniques are probably projection-based stereo optical systems inwhich axes of the projection system and imaging system are notcoaxial. These include the light-section method [2], the moiré
method [3], and the fringe projection method [4–6]. These
stereo optical methods, however, have difficulty measuringinside deep holes or nearby large step heights. In these cases,
uniaxial 3D profilometry methods are more practical.
An alternative approach to surface profilometry is uniaxial
fringe projection. While a stereo fringe projection system uses
the estimated fringe phases to infer the surface height, uniaxial
3D profilometry uses the change in fringe contrast as a functionof defocus to detect height information. Neil et al. used phaseshifting with fringe projection methods to perform optical sectioning and determined the change in fringe contrast with focus[7]. Takeda et al. proposed to use the contrast dependence on
focus to perform 3D profilometry for opaque samples [8]. A 3D
profilometry based on this principle was performed [9,10].
All of these fringe projection techniques need to collectmultiple measurements in order to measure the surface profile.
One modified approach uses the three colors (R, G, B) in a color
1559-128X/20/061654-06 Journal © 2020 Optical Society of America

camera to obtain three phase shift images simultaneously, fromwhich one can estimate the surface profile in a snapshot [11].
Samples with spatially varying color, however, will cause artifacts in the measurements, so we propose an alternate techniquethat instead makes use of polarization instead of color to encode
the contrast information. Using polarized light thus provides analternative method for colored samples. The polarization-based
approach integrates a polarization camera [12,13] into a uniaxial
fringe projection system, so that the four polarization orientations captured simultaneously by the camera allow snapshot 3D
profile measurements from four phase shift images.
2. EXPERIMENTAL SETUP AND
MEASUREMENT PRINCIPLES
The system measures the contrast of the light reflected froma sample using a spatially varying polarization pattern and apolarization camera. Figure 1 shows the experimental setup,consisting of a uniaxial projection system and an imaging
system. The polarization camera in the imaging system is composed of a micropolarizer array and detector array, where themicropolarizers are oriented at the four azimuth angles of 0◦ ,45◦ , 90◦ , and 135◦ .
The projection system produces the spatially varying polarization pattern using a quarter-wave plate (QWP) and a spatial
light modulator (SLM). Starting from the light source, laser
light is scattered by a rotating diffuser and collimated by lens
L1. The rotating diffuser reduces speckle noise in the detected
image, while L1 produces a collimated beam diameter matchedto the SLM and objective lens L3. The collimated light is linearlypolarized by polarizer LP and directed toward the SLM by the
non-polarizing beam splitter.

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Projection system

1655

Imaging system
Polarization Camera

Light Source

Detector array
Diffuser
L1

L4

Micropolarizer
array

LP
QWP

NPBS

NPBS

SLM

Beam dump

L2
Beam dump

L3

SampleFig. 1.

Experimental setup.

Step 2(From Light source)

Sin

Step 1

LP

SLM

QWP

NPBS

Sout

Step 3
Step 4(To Sample)

Beam dump

Fig. 2. Optical layout used to generate a spatially varying linear
polarization pattern.

Figure 2 shows a closeup of the beam near the SLM, showing
the polarization state of the beam before and after the SLM. The
linearly polarized light (step 1) is converted into circularly polarized light (step 2) by the QWP oriented at 45◦ to the beam’s
linear polarization angle. The circularly polarized light incident on the SLM is converted into spatially varying ellipticallypolarized light (step 3) by varying the phase delay across theSLM. As the return beam passes through the QWP again, theelliptical states are converted back into linear polarization, but
this time with spatially varying orientation angle (step 4). This
linear polarization pattern is projected on the sample by lenses
L2 and L3.

Fig. 3. (a) Linear polarization pattern incident on and reflectedby the sample. The arrow pattern indicates the local polarization
direction. (b) Relative intensity transmitted by each of the four micropolarizer orientations along the indicated cross-section location on thedetector array.

The polarization pattern reflected from the sample is imaged
onto the polarization camera’s detector array through lenses L3and L4. Figure 3 shows how the spatially varying linear polarization pattern incident on the micropolarizer array generates a
set of four phase-shifted fringe patterns through each of the four
micropolarizer orientations in the array.
We can characterize the linear polarization pattern (step 4of Figure 2) 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 turn

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gives the output Stokes vector Sout projected onto the sample
given byφ (x , y ) = tan−1

I45 − I135
I0 − I90



,

(5)

Sout (x , y ) = QWP(−45◦ ) · SLM(φ(x , y ), 0◦ ) · QWP(45◦ ) · LP(0◦ ) · Sin


 
 

rs + r p rs − r p
0
0
100 0
1100
1 0 00
1

1
r
+
r
r
−
r
0
0
0
0
0
−1
1
1
0
0
0


0 0 0 1



p s
p

 s
√
√
=
0
2 r s r p cos{φ(x , y )} 2 r s r p sin{φ(x , y )}   0 0 1 0  2  0 0 0 0   0 
0 0 1 0 0
√
√
0
010 0
0000
0−1 0 000 −2 r s r p sin{φ(x , y )} 2 r s r p cos{φ(x , y )}rs + r p1
rs − r p,√= 
2  2√ r s r p sin{φ(x , y )} 
2 r s r p cos{φ(x , y )}(1)
where φ(x , y ), r s , and r p represent the phase applied to theSLM, and the reflectances of s - and p-polarized light from theSLM. If we substitute r s = r p = 1 in Eq. (1), Sout becomes
1
 cos{φ(x , y )} .
Sout = sin{φ(x , y )} 
0where I0 , I45 , I90 , and I135 are
I0 = I (x , y , z, 0◦ ) , I45 = I (x , y , z, 45◦ ) ,I90 = I (x , y , z, 90◦ ) , I135 = I (x , y , z, 135◦ ) .

(2)From Eq. (2), we see that the azimuthal angle of linear polarization, given by tan−1 (S2 /S1 ), is equal to the phase φ(x , y ) of
the SLM. Thus the spatially varying angle of polarization is controlled simply by adjusting the phase φ(x , y ) at each column
across the SLM.
The light intensity I (x , y , ψ) detected by the polarization
camera is given by

Equation (4) shows that the contrast γ (x , y , z) can bemeasured in a snapshot from the four phase-shifted images.
For a non-polarizing sample, the contrast as a function of thesample surface’s distance from focus takes the form of a Bessel
function that can be approximated as a Gaussian function [10].
The contrast is maximum when the height of the sample is at
the best focus position of the optical system. During systemcalibration, we can fit the parameters of a Gaussian function toheight z as

I (x , y , ψ) = dT P (ψ) · Sout




1
cos (2ψ)sin (2ψ)0
1
2
  cos (2ψ)
1cos (2ψ)cos (2ψ) sin (2ψ) 0   cos {φ (x , y )} 
1000 
= sin (2ψ) cos (2ψ) sin (2ψ)
0   sin {φ (x , y )} 
sin2 (2ψ)
2
0
0
0
0
0
=

1
{1 + γ0 (x , y ) cos (φ (x , y ) − 2ψ)} ,2

where P (ψ) is the Mueller matrix of a micropolarizer oriented
at angle ψ ∈ {0◦ , 45◦ , 90◦ , 135◦ }, and γ0 (x , y ) is the ideal
contrast, defined by
γ0 (x , y ) =2r s r p.
r s2 + r 2p

From Eq. (3), we can obtain four phase-shifted patterns atonce from the detected intensity by the polarization camera by
separately analyzing the images behind the four micropolarizer
orientations. The contrast γ (x , y , z) and the phase φ(x , y , z)
are calculated by applying a four-step phase-shifting techniqueto the measured intensities:p
2 (I0 − I90 )2 + (I45 − I135 )2,(4)
γ (x , y , z) =I0 + I45 + I90 + I135

(3)

 !
1 z − z0 2
γ (x , y , z) = γ0 (x , y ) exp −
.
2
σ

(6)

Inverting Eq. (6) gives the surface height as a function ofcontrast:v!
uu
γ0 (x , y )tz (x , y ) = z0 + σ 2 ln(z ≥ z0 ) , (7)γ x , y, zs

γ0 (x , y )z (x , y ) = z0 − σ 2 ln
γ (x , y , z)(z ≤ z0 ) .

(8)

By measuring the contrast distribution using a planar reference sample in advance, we can calibrate parameters γ0 (x , y ),z0 , and σ in Eq. (6). For calibration, the reference sample should

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Fig. 6. (a) Projected image on the sample captured by the ordinary
camera (without micropolarizers). (b) Projected image on the sample
captured by the ordinary camera viewing the scene through the linear
polarizer.

where F is the F -number of the optical system, N is the numberof fringe patterns per millimeter, and M(= 3.832 ...) is the firstzero of the Bessel function J 1 .
3. EXPERIMENTAL RESULTS

Fig. 4. (a),(b) Wrapped phase map of the SLM pattern and its
cross-sectional view. (c),(d) Wrapped phase map and cross-section
view, measured by the polarization camera using Eq. (5), for a flat
reference sample.

produce a contrast profile (change with height that is proportional to that of the measurement sample, and which sharesthe measurement sample’s polarization characteristics). These
conditions are easily satisfied if the reference and measurement
sample surfaces are made of the same material, but this is not
the primary requirement. Note that the contrast profile is symmetric about z0 , so that this method cannot by itself distinguishbetween heights in the range of z ≥ z0 from those in z ≤ z0
using Eq. (7) or Eq. (8) alone. However, if we set up the measurement such that we ensure the sample heights only occur in thepositive range for z − z0 , then this is not a problem.
Because all four of the phase measurements needed for thecontrast estimate are obtained in a snapshot, this method canmeasure three-dimensional profiles in a single shot. Because
the optical transfer function of a fringe projection system isexpressed in the form of the Bessel function [8], the measurable
range of z (mm) is given by
0 ≤ z − z0 ≤orFig. 5.

−M·F(z ≥ z0 )π·N

M·F≤ z − z0 ≤ 0 (z ≤ z0 ) ,
π·N

(9)

In order to demonstrate the system, we performed two experiments on reference samples. For the experiments, we choselens focal lengths for L1, L2, L3, and L4 of 30 mm, 100 mm,
20 mm, and 200 mm. A f -number of the optical system is aboutf /15. The light source is a 532 nm Nd-YAG laser. The SLM isa 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 grid of 2.5 µm pixels. Therefore, thelateral range of the system is 0.51 mm × 0.39 mm.
For our first experiment, we measured a plane mirror as asample in order to validate the grating patterns detected by thepolarization camera. Figure 4 shows the phase map of the SLM
and the result of reshaping the phases of the fringes reflectedfrom the plane mirror, using Eq. (5). We confirmed the phasechanges linearly from −180◦ to 180◦ along the y-direction. The
result means that the angle of polarization changes linearly along
the y-direction. The phase map is used to generate the linear
polarization pattern on the sample to obtain the proper fourkinds of fringe patterns.
Figure 5 shows the image collected by the polarization camera, as seen by each of the four orientations of micropolarizer
pixels. We can see clearly phase shift between each of the four
images. Figure 6 shows the projected image captured by theordinary camera has a uniform distribution, but we can see
the fringe pattern from the projected image captured by theordinary camera through the linear polarizer.
Next, we measured the contrast of a plane mirror while theplane mirror was translated in 0.2 mm intervals by a z-axis stage.
Figure 7(a) shows the measurement results of the contrast fitted
by a Gauss function, and Fig. 7(b) shows the calibrated heightof the z-axis stage and versus the height estimated using Eq. (7).

Images detected for each polarizer orientation in the polarization camera.

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Fig. 7. (a) Measured contrast as a function of the sample height, and the fitted Gauss function (red curve). (b) Relationship between the sample
height and the height estimated by the fitted Gauss function.

(b) CrossC
section(AA-B)(a) Stepped sample
s

4
4.0
0 ms

mov
ving

Z(mm)

3
3.0

33 ms

66 ms

2
2.0

M (mirror)
M1

99 ms

M2 (mirro
or)

1
1.0

Z

0
0.0

X

0.2

0

0.4

X(mm)

(c)

(d)

(e)

(f)

Fig. 8. Results of the real-time 3D profile measurements of the step sample: (a) the stepped sample. (b) Cross section of the measured profile at fourtime points. The vertical axis gives the surface height for time t = 66 ms, and the other time points are vertically offset for clarity. (c)–(f ) 3D profile ofthe step sample at time points 0 ms, 33 ms, 66 ms, and 99 ms (see Visualization 1).

Research Article
For heights within the range of 0.3–2.4 mm, the average heighterror is 0.04 mm. Therefore, the depth range of the system is
2.1 mm, and the resolving power (the number of resolvableelements) along the z-direction is 2.1 mm/0.04 mm = 52.
Next, we measured the 3D profile of the moving steppedsample to demonstrate real-time 3D profilometry, the results ofwhich are shown in Fig. 8. During the experiment, the steppedsample 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 0.48 mm × 0.28 mm. The thickness of
mirrors M1 and M2 is 1.0 mm. The height is calculated fromthe measured contrast using the calibrated curve of the reference
plane mirror (Fig. 7).
4. CONCLUSION
The real-time uniaxial 3D profilometry system we demonstrate here fulfills a need for high-precision, high-speed, andhigh-resolution measurements of the shapes of objects in 3D.
A previous method for doing this employed color diversity tocapture three fringe patterns in a snapshot, but this can cause
artifacts for objects with spatially varying color features. Here weuse a polarization camera and polarization pattern projection,
which is accurate for objects having spatially homogeneouspolarization properties. This adds another technique to the toolbox of engineers designing profilometry systems, and delivers
3D data at a frame rate limited only by the speed of the camera.
The uniaxial construction of the measurement gives an advantage for measuring deep holes and large step heights—featuresthat are difficult for stereo projection systems to measure.
The system we demonstrate here is suitable for samples
with height variations in the millimeter range, but this canbe adjusted for larger height ranges by using larger projection
optics, or for smaller height features by using a higher numericalaperture objective lens. For a given SNR, the height resolving
power (calculated as 52 above) remains the same regardless ofthe projection optics. This system is well-suited to evaluate
shapes of a sample composed of a single material. However,the problem of this system is that it cannot measure a sample

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composed of materials with different depolarization because thecontrast depends on the depolarization.
Disclosures. The authors declare no conflicts of interest.
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