THREE-DIMENSIONAL DISPLACEMENT ANALYSIS BY WINDOWED PHASESHIFTING DIGITAL HOLOGRAPHIC INTERFEROMETRY
Yoshiharu Morimoto, Toru Matui and Motoharu Fujigaki
Department of Opto-Mechatronics, Faculty of Systems Engineering,
Wakayama University
Sakaedani, Wakayama 640-8510, Japan
[email protected]
ABSTRACT
Phase-shifting digital holography is a new method to measure the displacement distribution on the surface of an object. The
authors previously proposed the windowed phase-shifting digital holographic interferometry (Windowed PSDHI). The method
provides accurate displacement distribution decreasing the effect of speckle patterns. In this paper, the method is extended to
analyze three-dimensional displacement components. Laser beams with spherical waves are used to miniaturize the
equipment for practical use. In order to calibrate the directions of incident angles and observation angles on an object, a
calibration method with a reference plane is proposed. The reference plane is installed on an XYZ three-axis piezo stage. The
sensitivity vector components at each point on the reference plane are obtained from an experiment for calibration.
Deformation measurement using spherical waves can be realized with this calibration method. Theoretical treatment and
experimental results of three-dimensional displacement measurement using this method are shown.
Introduction
Digital holography using a CCD sensor and digital image processing is useful to reconstruct the three-dimensional shape of an
object [1]. The interference of the object beam and reference beam is recorded on the CCD sensor as a hologram. The
complex amplitude of each point of the object is analyzed from the phase-shifted holograms obtained by shifting the phase of
the reference beam. The reconstructed image can be calculated from the complex amplitudes of the hologram [2, 3]. The
reconstruction process of the objects is fast and accurate because the reconstruction is performed by computer calculation
without developing any photographic plate. Phase-shifting digital holographic interferometry is possible to measure shapes
and displacements of objects quantitatively by analyzing the phase information of laser beam waves. Speckle patterns in the
holography, however, provide noise, and the accuracy of displacement measurement is not so good. The authors developed
windowed phase-shifting digital holographic interferometry (Windowed PSDHI) which provides very accurate results by
decreasing the effect of speckle noise [4,5]. The most accurate result by the Windowed PSDHI for analyzing small
displacement in the authors experiment, 88 picometer resolution was obtained. In this paper, the method is extended to
analyze three-dimensional displacement components.
The authors proposed a method of simultaneous measurement of in-plane and out-of-plane displacements using the two
beam illuminations [6]. Using this method, a measurement system was developed [7, 8]. In order to measure threedimensional displacement components, three-directional-illumination method or three-directional-observation method is usually
employed. In this paper, a three-directional illumination method is proposed. To miniaturize the equipment for practical use,
spherical waves of laser beams are used. However, if the spherical waves are used, incident angles are different for each
point and then each point has different sensitivity vectors. The incident angle for each point on an object is determined by the
three-dimensional positions of the point and the point source of the laser beam. It is, however, difficult to measure the incident
angle accurately. The authors propose a calibration method with a reference flat plane. The reference plane is installed on an
XYZ three-axis piezo stage which is movable in the XYZ three-axis directions with a very small amount. The reference plane
is moved in three directions with a very small amount. By calculating the each phase-difference between before and after
deformation using the digital holography, the parameter for the relationship between the displacement and the phasedifference can be obtained. Tabulation of parameters for each point helps to measure the displacement in high speed from the
phase-difference of a specimen. Displacement measurement using spherical waves can be realized with this calibration
method. Theoretical treatment and experimental result of three-dimensional displacement measurement using this method are
shown.
Windowed Phase-shifting Digital Holographic Interferometry (Windowed PSDHI)
In phase-shifting digital holography, a hologram is usually recorded on a CCD sensor instead of a photographic plate. The
complex amplitude of the object is analyzed from the phase-shifted holograms obtained by shifting the phase of the reference
beam. The reconstructed image can be calculated from the complex amplitudes of the hologram [2-3].
By calculating the Fresnel diffraction integral from the complex amplitudes g(X, Y) at the position (X, Y) on the CCD plane, the
complex amplitude u(x, y) of the reconstructed image at the position (x, y) on the reconstructed object surface being at the
distance R from the CCD plane is expressed as follows.
$!
- ik ( x 2 + y 2 ) * '! - ik (X 2 + Y 2 ) *
u ( x , y) = exp +
( F&exp +
( g ( X, Y ) #
2R
2R
!"
,+
)( !% ,+
)(
(1)
where k and F denote the wave number of the light and the operator of Fourier transform, respectively. The optical axis is
normal to the CCD plane and the origin goes through the center of the CCD plane. By calculating the intensities of these
complex amplitudes on the reconstructed object surface, a holographic reconstructed image is obtained in a short time by a
computer.
In phase-shifting digital holographic interferometry for measuring a displacement distribution of an object, displacement at
each point is obtained from the phase-difference between digital holograms recorded before and after deformation. Figure 1
shows an experimental setup of the digital holographic interferometry. Small displacement of a cantilever is analyzed. Figure
2(a) shows the analyzed phase distribution. Figure 3(a) shows the displacement distribution along the centerline. The standard
deviation of the errors from the theoretical displacement is 16.4nm.
Holograms and reconstructed images have speckle noise and they provide large errors in the calculation of displacement
analysis. In order to reduce the effect of speckle noise, the authors proposed a novel method using many windowed
holograms [4-5].
Figure 1 Optical system for phase-shifting digital holography
(a) From whole hologram (n=1) (b) From divided holograms(n=16)
Figure 2 Phase-difference distributions obtained by digital holographic interferometry
(a) by whole hologram (SD=16.4nm)
(b) by 16 windowed holograms (SD=1.95nm)
Figure 3 Displacement distribution along lines A and B shown
In holography, any part of a hologram has the optical information of the whole reconstructed image. By using the feature of
hologram, the hologram is divided into some parts by superposing many different windows as shown in Fig. 4(a). In this case,
16 windowed holograms are used. Figure 4(b) shows the reconstructed object images from the 16 holograms, respectively.
The phase-difference values at the same reconstructed point obtained from any other different part of the hologram should be
the same. However, the values are not the same because of speckle noise which is different according to the hologram with a
different position. If there is speckle noise, the phase-difference with higher intensity at a reconstructed point is more reliable.
Therefore, the phase-difference at the same point is calculated by averaging the phase-differences obtained from all the
windowed holograms by considering the weight of the intensity. It provided the displacement distribution with high-resolution.
Figure 2(b) shows the analyzed phase-difference distribution obtained from 16 windowed holograms shown in Fig. 4(a). Figure
3(b) shows the displacement distribution along the centerline. The standard deviation of the errors from the theoretical
displacement is 1.95nm.
The effect of the number n of the windowed holograms is checked for flat plate movement. The best standard deviation of the
errors is 88 picometer when n=1024.
(a) Divided holograms
(b) Reconstructed object images
Figure 4 Divided holograms and reconstructed object images
Relationship between Phase-differences and Displacement Components
The phase-difference for a unit displacement is depending on the position of the object. When the positional relationship is
expressed in Fig. 5, the equation at an point P on an object is expressed as shown in Eq. (2) [9].
"# = e ! d
(2)
where, e is the sensitivity vector which depends on the half of the angle between incident angle θi and observation angle θo of
the point P, and d is the displacement vector for the point P, and Δφ is the phase-difference resulted from the displacement.
Figure 5 Positional relationship between light source, object and observation point
The displacement vector d and the sensitivity vector e have each component of the x, y, and z directions for the incident light.
Then they are expressed as follows;
(' = &$ ex
%
ey
&d x #
ez # $$d y !! = ex d x + e y d y + ez d z
!"
$d !
% z"
(3)
When an object is exposed from three different directions, the number of parameters of the sensitivity vector components
increases, and Eq. (3) can be extended as Eq. (4);
& ('1 # & e1x
$
! $
$('2 ! = $e2 x
$ (' ! $e
% 3 " %$ 3 x
e1 y
e2 y
e3 y
e1z #
!
e2 z !
e3 z !"!
&d x #
$ !
$d y !
$d !
% z"
(4)
where the suffixes 1, 2 and 3 show corresponding each incident lights, respectively.
When each component of the matrix on the right side of Eq. (4) is specified, the displacement component dx dy and dz can be
obtained from phase-difference Δφ㸡Δφ2 and Δφ3 for each incident light, respectively.
Calibration Method Using Reference Plane
In this chapter, a calibration method is proposed using reference plane to obtain values of each component of the matrix e on
the right side of Eq. (4). The reference plane is installed on the three-axis PZT stage as shown in Fig. 6 and it can be
translated slightly in the three directions of x, y and z. The reference plane is set at the same position as a target object and the
spherical waves are exposed from three directions.
Let us discuss a point on the reference plane. When the reference plane is moved by Δx toward the x direction, the matrix
components ex, e2x and e3x can be obtained as Δφ1/Δx㸡Δ φ2/Δx and Δ φ3/Δx respectively by specifying the phase-difference
Δφ1, Δφ2 and Δ φ3 for each light source. In the same way, all the matrix components can be obtained by moving the reference
plane toward the y direction and the z direction. By repeating the procedures to average the each component, accurate matrix
components can be obtained.
Because the matrix components are calculated for each pixel of the reconstructed image, the displacement can be easily
obtained from phase-difference even though it is spherical wave.
Figure 6 Spherical wave from three light sources and reference plane placed on three-axis PZT stage
Let the matrix on the right side of Eq. (4) be S and each component of the inverse matrix is expressed as Eq. (5). Where, all
sensitivity vectors for each direction of the each incident light are independent. Then S is a normal matrix. Because the normal
matrix makes the inverse matrix, S-1 is the inverse matrix of S and Eq. (4) can be expressed as Eq. (5).
& f 1x
$
S '1 = $ f 1 y
$% f 1z
& d x # & f 1x
$ ! $
$d y ! = $ f 2 x
$d ! $ f 3x
% z" %
f 2x
f2y
f 2z
f1y
f2y
f3y
f 3x #
!
f3y !
f 3 z !"
f 1z #
!
f 2z !
f 3 z !"
(5)
& ('1 #
$
!
$(' 2 !
$ (' !
% 3"
(6)
The results are tabulated for each pixel.
Experiment for Principle Confirmation
Experimental Setup
The optical system used in the experiment is shown in Fig. 7. For the light source, a He-Ne laser with output of 8 mW and
wave length of 632.8 µm is used. A digital interface CCD camera with the pixel size 4.65 mm x 4.65 mm is used. The number
of pixels of the captured image is 960 x 960 pixels and gray-scale is 256 steps. The distance between the target object and the
CCD camera is 318 mm. The light source is divided into a reference beam and three object beams 1, 2 and 3 by a beam
splitter and two half mirrors as shown in Fig. 7.
Figure 7. Optical setup for three-dimensional displacement measurement
Calibration
Figure 8 shows a reference plane set on the three-axis PZT stage. The reference plane is placed at the positon of the object
in Fig. 7. The material of the reference plane is anodized aluminum. In this experiment, 100 nm of displacement is applied to
each direction of the x, y and z directions. Four images of digital holograms are captured several times while the reference
plane is being moved in each direction. The phase-shifted digital holograms of the reference plane are shown in Fig. 9 as an
example. The holograms are exposed by the object beam 1 and the reference plane is moved toward the x-direction. The
phase distribution is calculated from these four digital holograms. The phase-difference between before t and after movement
is calculated. The sensitivity vector components for each direction and each incident light are calculated from the applied
displacement. The distributions of the sensitivity vector components of the x, y and z-directions by the object beam 1 are shown
in Fig. 10.
The phase-difference distribution is obtained by calculating the sensitivity vector components using the Windowed PSDHI with
64 windows. Usually the sensitivity vector components change smoothly. The sensitivity vector components obtained in our
experiment, however, do not change smoothly even though the Windowed PSDHI is performed. To solve this problem,
smoothing process is applied to the sensitivity vector components in the range of 21 x 21 pixels. The obtained sensitivity
vector component values on a line before and after smoothing process are shown in Fig. 11. Then smooth sensitivity vector
components are obtained. (࡝ࡏࡣࡼࡗࡂࡡ࠾)
As a result, calibration for displacement measurement is performed in the optical system and nine sensitivity components at
each pixel are obtained. Tabulation of the components for each point is performed.
Figure 8 Reference plane
Figure 9 Phase-shifting digital holograms of reference plane
(a) x-direction (b) y-direction (c) z-direction
Figure 10 Sensitivity vector maps of object beam 1
Figure 11 Smoothing of sensitivity vector map
Experiment of Displacement Measurement
A rotating plate is used as a specimen as shown in Fig. 12. Some concentric circles are drawn on the plate. Displacement
distribution measurement is performed when the plate is rotated by 0.01 degree.
The reconstructed images from the digital holograms obtained by exposing from the object beam 1, 2 and 3 are shown in Fig.
13(a), (b) and (c), respectively. The phase-difference distributions obtained using the object beam1, 2 and 3 between before
and after rotating are shown in Fig. 14(a), (b) and (c), respectively. Because a phase-difference distribution involves speckle
noise, the Windowed PSDHI with 16 windows is performed. The phase-difference distribution after phase unwrapping is shown
in Fig. 15.
Three-dimensional displacement distributions are calculated by using the sensitivity vectors obtained in the previous section.
Each calculated displacement distribution is shown in Fig. 16. The displacement distribution along the Line A and the Line B in
Fig. 16 are shown in Fig. 17(a) and (b), respectively. As the result, the displacement for the x-direction is 6.24 mm and the
error is only 0.18 µm comparing with the theoretical value 6.06 µm. For the y-direction, it is 6.63 µm and the error is only 0.16
µm comparing with the theoretical value 6.47 µm. In this case, the z-directional displacement is zero theoretically and the
experimental results in Fig. 17 show almost zero. As the results, the three-dimensional displacement distributions are obtained
accurately.
Figure 12 Specimen set on rotating stage
(a) by object beam 1 (b) by object beam 2 (c) by object beam 3
Figure 13 Reconstructed image
(a) by object beam 1 (b) by object beam 2 (c) by object beam 3
Figure 14 Phase-difference of specimen
(a) by object beam 1 (b) by object beam 2 (c) by object beam 3
Figure 15 Unwrapped phase-difference of Fig. 14
(a) x-directional
(b) y-directional
(c) z-directional
Figure 16 Displacement distributions
(a) Along line A
(b) Along line B
Figure 17 Displacement distribution along line A and line B in Fig. 16
Conclusions
In phase-shifting digital holography, the three- dimensional windowed phase-shifting digital holographic interferometry (3DWindowed PSDHI) using spherical waves for object waves was proposed. The method to obtain the sensitivity components for
each pixel using the reference plane was proposed. Experimental results of deformation measurement using this method were
shown. It was confirmed that the proposed method is effective.
Acknowledgments
A part of this study was supported by a grant aided project for creation of new regional industry of Kansai Bureau of Economy
Trade and Industry. The authors also appreciate the support by Mr. T. Kita and Mr. S. Okazawa, Graduate School of
Wakayama University, and Mr. A. Kitagawa and Mr. M. Nakatani of Hitachi Zosen Corporation.
References
1. Schnars, U. and Jüptner, W., Digital Holography, Springer, (2005).
2. Yamaguchi, I. and Zhang, T., Phase-shifting digital holography, Optics Letters, 22(16), pp.1268-1270, (1997).
3. Zhang, T. and Yamaguchi, I., Three-dimensional Microscopy with Phase-shifting Digital Holography, Optics Letters, 23-15,
1221, (1998).
4. Morimoto, Y., Nomura, T., Fujigaki, M. Yoneyama, S. and Takahashi, I., Deformation Measurement by Phase-shifting Digital
Holography, Experimental Mechanics, 45(1), 65-70, (2005).
5. Morimoto, Y. , Matui, T., Fujigaki, M. and Kawagishi, N., Subnanometer Displacement Measurement by Averaging of Phasedifference in Windowed Digital Holographic Interferometry, Optical Engineering, Vol. 46, No. 2, (2007).
6. Okazawa, S., Fujigaki, M., Morimoto, Y. and Matui, T., Simultaneous Measurement of Out-of-plane and In-plane
Displacements by Phase-Shifting Digital Holographic Interferometry, Applied Mechanics and Materials, 3-4, 223, (2005).
7. Fujigaki, M., Matui, T., Morimoto, Y., Kita, T., Nakatani, M. and Kitagawa A., Development of Real-Time Displacement
Measurement System Using Phase-Shifting Digital Holography, Proceedings of the 5th International Conference on
Mechanics of Time Dependent Materials (MTDM05), 160-163, (2005).
8. Morimoto, Y., Matui, T., Fujigaki, M. and Kawagishi, N., Effect of Weight in Averaging of Phases on Accuracy in Windowed
Digital Holographic Interferometry for Pico-meter Displacement Measurement, Proc. of SPIE Vol. 6049, 60490D(CDROM), (2005) .
9. Fujigaki, M., Kita, T., Okazawa, S., Matui, T. and Morimoto, Y., CALIBRATION METHOD WITH REFERENCE PLANE FOR
PHASE-SHIFTING DIGITAL HOLOGRAPHIC INTERFEROMETRY USING SPHERICAL WAVE, International Symposium
on Advanced Fluid/Solid Science and Technology in Experimental Mechanics, 11-14, Sapporo, Japan, (2006)