March 1, 2007 / Vol. 32, No.

5 / OPTICS LETTERS 481

Polarization imaging of a 3D object by use of

on-axis phase-shifting digital holography
Takanori Nomura
Department of Opto-Mechatronics, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan

Bahram Javidi
Electrical and Computer Engineering Department, University of Connecticut, 371 Fairfield Road, Unit 2157,
Storrs, Connecticut 06269-2157, USA

Shinji Murata, Eiji Nitanai, and Takuhisa Numata

Department of Opto-Mechatronics, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan
Received August 10, 2006; revised November 30, 2006; accepted December 2, 2006;
posted December 5, 2006 (Doc. ID 73969); published February 2, 2007
A polarimetric imaging method of a 3D object by use of on-axis phase-shifting digital holography is pre-
sented. The polarimetric image results from a combination of two kinds of holographic imaging using or-
thogonal polarized reference waves. Experimental demonstration of a 3D polarimetric imaging is presented.
© 2007 Optical Society of America
OCIS codes: 200.4740, 090.0090, 120.5410, 110.6880.

Digital holography is a useful technique for recording consisting of opposite polarization states, enable us
the fully complex field of a wavefront.1–4 In line with to reconstruct a 3D object with polarimetric informa-
advances in imaging devices such as CCDs, digital tion.
holography is accessible. A phase-shifting technique2 Let uoh共x , y兲 and uov共x , y兲 denote the horizontal and
is mandatory for effective utilization of the number of vertical components of the electric field of an object
pixels and pixel size of an imaging device. Digital ho- wave, respectively. Let urh共x , y兲 and urv共x , y兲 denote
lography has been used for lots of applications, in- those of the reference wave, respectively. Using the
cluding encryption5,6 and 3D object recognition.7,8 On horizontal and the vertical amplitude distributions of
the other hand, the polarization information provides the object wave aoh共x , y兲 and aov共x , y兲, the horizontal
important information about the surface of the mate- and the vertical phase distributions of the object
rial, birefringence of an anisotropic medium, or pho- wave, ␾oh共x , y兲 and ␾ov共x , y兲, respectively, and those of
toelastic effect.9–11 In polarimetric imaging by use of the reference wave, arh共x , y兲, arv共x , y兲, ␾rh共x , y兲, and
digital holography,10 the polarization state (Jones ␾rv共x , y兲, we write each component of the two waves
vector) of a 2D transparent object has been obtained as
using two off-axis reference waves with orthogonal
polarizations. Because reconstruction of a 3D object uoh共x,y兲 = aoh共x,y兲exp关i␾oh共x,y兲兴, 共1兲
is one of the most important aspects of holography, it
is useful if a polarimetric imaging system of a 3D ob- urh共x,y兲 = arh共x,y兲exp关i␾rh共x,y兲 + ␣兴, 共2兲
ject is accomplished. Namely, we know both the pola-
rimetric information and the shape of a 3D object si-
multaneously. One of the authors has proposed 3D uov共x,y兲 = aov共x,y兲exp关i␾ov共x,y兲兴, 共3兲
polarimetric imaging based on integral imaging.12
However, to the best of our knowledge digital holo- urv共x,y兲 = arv共x,y兲exp关i␾rv共x,y兲 + ␣兴, 共4兲
graphic 3D imaging with polarization has not been
reported. In this Letter, to realize polarimetric imag-
ing of a 3D object, we propose the use of two kinds of where the parameter ␣ denotes the phase-shifting
digital holograms obtained by orthogonal polarized quantity of the reference wave. From here we explain
reference waves. the vertical components case for simplicity. We have
We have proposed a phase-shifting digital hologra- a hologram written as
phy system with a phase difference between orthogo-
Iv共x,y, ␣兲 = aov
2 2
共x,y兲 + arv 共x,y兲 + 2aov共x,y兲arv共x,y兲
nal polarizations.13 Using the proposed technique, we
can obtain the polarimetric information of the object ⫻ cos关␾ov共x,y兲 − 兵␾rv共x,y兲 + ␣其兴. 共5兲
by recording the phase-shifted digital holograms. In
this Letter, we make use of the polarimetric informa-
The holograms using 0 and ␲ / 2 phase-shifted refer-
tion directly. Therefore we propose a 3D polarimetric
ence waves are given by
imaging technique by use of digital holography. Two
reconstructed objects, one consisting of a reference 2
Iv共x,y,0兲 = aov 2
共x,y兲 + arv 共x,y兲 + 2aov共x,y兲arv共x,y兲
wave with a vertical polarization state and an object
wave with a horizontal polarization state, the other ⫻ cos兵␾ov共x,y兲 − ␾rv共x,y兲其, 共6兲
0146-9592/07/050481-3/$15.00 © 2007 Optical Society of America
482 OPTICS LETTERS / Vol. 32, No. 5 / March 1, 2007

Iv共x,y, ␲/2兲 = aov

2 2
共x,y兲 + arv 共x,y兲 + 2aov共x,y兲arv共x,y兲
⫻ sin兵␾ov共x,y兲 − ␾rv共x,y兲其. 共7兲
If we know the two intensity distributions Irv共x , y兲
and Iov共x , y兲 given by
2
Irv共x,y兲 = arv 共x,y兲, 共8兲

2
Iov共x,y兲 = aov 共x,y兲, 共9兲
then the fully complex information of the object wave
can be obtained by
Fig. 1. Optical setup for polarimetric imaging by use of
aov共x,y兲 = 冑Iov共x,y兲, 共10兲 digital holography: SF1, SF2, spatial filters; CL1, CL2, col-
limating lenses; BSl, BS2, beam splitters; M1, M2, mirrors;
O, object; A, analyzer; HWP, half-wave plate; QWP,
Iv共x,y, ␲/2兲 − Iov共x,y兲 − Irv共x,y兲 quarter-wave plate.
␾ov共x,y兲 = tan−1
Iv共x,y,0兲 − Iov共x,y兲 − Irv共x,y兲
+ ␾rv共x,y兲. 共11兲
Since the constant phase distribution ␾rv共x , y兲 of the
reference wave can be ignored, Eq. (11) can be consid-
ered as the object phase distribution. Finally we can
obtain the vertical component of the object wave. The
horizontal component can be obtained in the same
way.
If we have both horizontal and vertical components
of an object wave, we can obtain the Stokes param-
eters of the object. Using the horizontal and the ver-
tical amplitude distributions of the reconstructed ob-
Fig. 2. (a) Randomly polarized object and (b) a polarimet-
ject, Aoh共X , Y兲 and Aov共X , Y兲, and the horizontal and ric object.
the vertical phase distributions of the reconstructed
object, ⌽oh共X , Y兲 and ⌽ov共X , Y兲, the Stokes vector is set along the fast axis of the QWP, we can record
given by Iv共x , y , 0兲. The holograms contain the vertical polari-

冤 冥冤 冥
2 2 metric information of the object (O). If either an ob-
S0共X,Y兲 Aoh 共X,Y兲 + Aov 共X,Y兲
ject wave or a reference wave is blocked, we can ob-
S1共X,Y兲 2
Aoh 2
共X,Y兲 − Aov 共X,Y兲 tain the intensity distributions of the remaining
S= = , wave.
S2共X,Y兲 2Aoh共X,Y兲Aov共X,Y兲cos ␧共X,Y兲
If we rotate the HWP to set the polarization state
S3共X,Y兲 2Aoh共X,Y兲Aov共X,Y兲sin ␧共X,Y兲 of the beam horizontal, we can obtain horizontal po-
共12兲 larimetric information of the object. In this case, the
transmission axis of the analyzer is set along the
where ␧共X , Y兲 denotes the phase-difference distribu- slow axis of the QWP.
tions between a horizontal and a vertical component To confirm the proposed method, preliminary opti-
given by cal experimental results are given. A He– Ne laser
␧共X,Y兲 = ⌽oh共X,Y兲 − ⌽ov共X,Y兲. 共13兲 (with a wavelength of 632.8 nm) that is in a vertical
polarization state is used as a coherent light source.
This vector gives the polarimetric information of the The 3D object, the die shown in Fig. 2, is used for
object. both the randomly polarized and the polarimetric
The optical setup that records the digital hologram cases. The size of the die is 8 mm⫻ 8 mm⫻ 8 mm.
with polarization information is shown in Fig. 1. It is Two orthogonal polarizers are placed in front of the
assumed that the polarization state is vertical after polarimetric object. The arrows in Fig. 2(b) show the
passing through the half-wave plate (HWP). The direction of the transmission axes of the polarizers.
light is divided by a beam splitter 1 (BS1) into two The left polarizer has the horizontal transmission
waves. One is a reference wave and the other is an axis, and the right is vertical. The upper central por-
object wave. The reference wave passes through a tion of the two orthogonal polarizers overlap. On the
quarter-wave plate (QWP). The fast axis and the other hand, there are no polarizers in the lower cen-
QWP is set at an angle of 45° with respect to the hori- tral portion. The CCD camera has 1024⫻ 768 pixels
zontal axis. If we detect the reference beam along and 8 bits of gray level. The pixel size of the CCD is
with the fast and the slow axes of the QWP sepa- 4.65 ␮m ⫻ 4.65 ␮m. The distance from the die to the
rately, the phase difference in the two beams is equal CCD is 230 mm. The reconstructed objects are ob-
to ␲ / 2. To detect each beam, we use a rotatable ana- tained from the holograms using computational
lyzer (A). If the transmission axis of the analyzer is Fresnel diffraction integral. The gray-scale images
 March 1, 2007 / Vol. 32, No. 5 / OPTICS LETTERS 483

according to the Stokes parameters of the recon-

structed objects are shown in Figs. 3 and 4. Each fig-
ure is normalized so that black and white denote
minimum and maximum values, respectively. The
S0共x , y兲 images are shown in Figs. 3(a) and 4(a).
These are ordinal digital holographic reconstructed
objects. The S1共x , y兲 images are shown in Figs. 3(b)
and 4(b). The white and black denote the horizontal
and vertical components of polarization, respectively.
To clear the characteristics of the polarimetric object,
we show another image in Fig. 4(c) corresponding to
S1⬘ 共X , Y兲, defined by
S1⬘ 共X,Y兲 = Aov
2 2
共X,Y兲 − Aoh 共X,Y兲. 共14兲

In this figure, the white and black denote the vertical

and horizontal components of polarization, respec- Fig. 4. Gray-scale images according to the Stokes param-
tively. From Figs. 4(b) and 4(c), we learn that the im- eters (a) S0共X , Y兲, (b) S1共X , Y兲, (c) S1⬘ 共X , Y兲, (d) S2共X , Y兲, and
age represents the characteristics of the polarimetric (e) S3共X , Y兲 for a polarimetric object.
object. Figures 3(c), 3(d), 4(d), and 4(e) also make
suggestions about the polarimetric information of the objects. One consists of a reference wave with a ver-
object. These experimental results illustrate that 3D tical polarization state and an object wave with a
polarimetric imaging is possible. horizontal polarization state, the other consists of op-
In conclusion, we have proposed a method to obtain posite polarization states. We gave some experimen-
3D polarimetric imaging by use of digital holography. tal results of polarimetric imaging to confirm the
In the proposed method, we used two reconstructed method. The analysis of the polarimetric imaging is
under way.
This research is partially supported by the Japan
Society for Promotion of Science, Grant-in-Aid for
Scientific Research (C), 18560035, 2006. T. Nomura’s
e-mail address is nom@sys.wakayama-u.ac.jp.
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