Application of digital holography for polarization analysis
Masayuki Yokota
Department of Electronic Engineering, Gunma University,
1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan
Keywords: digital holography, polarization, photoelastic, inner stress, surface/deformation measurement
Abstract
Digital holography is applied to analyze the spatial distribution of polarization state of light transmitted through
anisotropic objects by successive hologram recording using polarization switching of the reference wave by an
optical fiber Faraday rotator (FFR) [1]. The distribution of polarization ellipse over a quarter wave plate surface is
analyzed for various azimuth of the plate. By inserting a polarizer into a part of the object wave after the object as
a reference area, a phase drift of the reference waves caused by polarization switching and/or environmental
disturbances can be evaluated and compensated for by using the reference area [2, 3]. A good agreement between
the experimental and theoretical values is observed. For an application, a photoelastic effect induced in a stressed
PMMA specimen is also investigated.
1. Introduction
Distribution of optical polarization state has been analyzed by means of a polarization imaging of a sample of
interest [4, 5]. The polarization imaging is also possible by holography using a polarized reference beam.
Especially, digital holography is suited for quantitative analysis of polarization distribution [6, 7]. The advantages
of digital holography over the other polarimetry are relatively simple optical system without any imaging lens
and rotating optics, and the adaptability to three-dimensional objects because of numerical focusing. The shape or
deformation of object surface can be detected by means of phase information of a reconstructed object wave in
digital holography. This is also advantage over a traditional photoelastic analysis.
In digital holography, a hologram is recorded by an imaging device such as CCD and the image is
numerically reconstructed by computing the diffraction integral. Therefore, the numerical values of amplitude
and phase of the image make it possible to achieve quantitative analysis. To construct a simpler configuration and
analysis, we have proposed a new method using an orthogonal linearly polarized single reference beam whose
orientation is controlled by the FFR. The introduction of the FFR realizes the single reference beam, compact,
and flexible optical setup without any mechanical movement such as rotation of a polarizer. We apply an off-axis
setup for successive hologram recording with polarization switching of the reference beam. The analysis of the
reconstructed image is also simpler because we need no coordinate transformation for comparing reconstructed
images corresponding to orthogonal polarizations. In addition, the influences of lens aberrations, tilt and optical
misalignment also remain the same for the orthogonal reference beams. It has been shown that the effect of phase
drift of the reference beam between two exposures can be cancelled by introducing a linearly polarized beam in
the object field.
In this paper, we investigated polarization states of the transmitted light from a QWP and an influence of
polarization dependence of a beam splitter in the system. Then, we applied the technique for investigation of a
stress-induced birefringence of PMMA specimen for application.
1
2. Experimental configuration
The experimental configuration of polarization recording is shown in Fig. 1. The light beam from a laser diode
(BlueSky Research PS026-00:  = 784.45 nm, Po = 10 mW) is launched into a single-mode-fiber coupler via an
optical isolator (OI) and two lenses (L1 and OL) to split into reference and object arms. Two fiber Faraday
rotators (FFR) and an optical fiber polarizer (Pol1) are inserted into the reference arm to generate orthogonal
polarization sequentially. The Faraday rotator consisting of an acrylic core in which a 3-m flint glass fiber coil
covered with a glass fiber tube is inserted. The glass fiber tube acts as a heat-insulator. The acrylic core is also
covered with a heat insulating glue and a modulation coil is wound around it. Two Faraday rotators achieve an
efficient modulation of polarization orientation with an efficiency of 32.1 deg/A. In the object arm a pair of fixed
Glan-Thompson polarizers (Pol2 and Pol3) are inserted to fix the polarization state and the illuminating intensity
for an object. These polarizers have an extinction ratio of 50 dB. The extinction ratio of the reference beam is
about 30 dB. A polarizer pol4 is partially inserted into an area of object wave after the object to introduce a
known polarization. The hologram is recorded in an off-axis geometry. The object beam is normally incident on
the CCD surface, whereas the reference beam has an incident angle of 1°.
The two beams are combined by a beam splitter BS and brought to interference. Two holograms are recorded
sequentially by changing the polarization orientation of the reference beam by the Faraday rotator with the coil
current of ±1.4 A. The CCD camera (IMPERX IPX-VGA120) whose pixel has an area of 7.4×7.4 m2 records a
hologram of 512×512 pixels with 8-bit resolution and maximum frame rate of 110 Hz. It requires about 0.1 s for
acquisition of two holograms with the orthogonal reference beams. The video data are stored in a frame grabber
(Matrox Meteor2CL) and sent to a PC for a numerical reconstruction.
Amp
FFR
Pol1
LD
M
OI
L1
OL
PC
L2
FC
L3
Pol3 Sample
CCD
Vertical polarization (⊥)
Pol2
Pol4
BS
Fig. 1 Experimental setup: LD, laser diode; OI, optical isolator; FC, fiber coupler; FFR, flint glass fiber Faraday
rotator; M, mirror; BS, beam splitter; L, lens; Pol, polarizer; Amp; current amplifier; PC, personal computer.
3. Polarization analysis
To confirm the principle for determination of polarization state, a quarter wave plate (QWP) was used for an
object. The incident beam was linearly polarized light whose orientation was 45° to the horizontal axis. The
polarization state of the transmitted light beam was analyzed for various clockwise orientation  of the axis of
QWP changed between 0° and 180° with 5° step. The object distance was 169 mm from the CCD surface.
2
B
B
(a)
Phase difference Δ(deg)
3 mm
Amplitude ratio (deg)
For analysis, a phase drift occurring in the polarization switching of reference beam induced a phase difference
offset. It causes a crucial error in determination of the polarization state. We applied the technique proposed by
Colomb et al. to compensate for the phase difference offset. To evaluate the offset value, a polarizer (Pol4) whose
orientation was 45° is inserted into a small area of the object beam behind the QWP to provide a reference area of
linear polarization state in which the phase difference of orthogonal components has to be 0°. Therefore, the
experimental value of phase difference obtained from the reference area was averaged over it and the mean value
was used for the offset value. This value was added to the derived phase difference value in the measured area of
QWP to provide the corrected phase difference.
: Theoretical
: Phase corrected
: Amplitude and phase corrected
70
60
50
40
30
A
A
0
60
90
120
150
180
50
0
–50
: Theoretical
: Phase corrected
–100
0
30
60
Principal axis (deg)
()
: Theoretical
: Phase corrected
: Amplitude and phase corrected
40
20
0
–20
30
60
90
120
150
Rotation angle  (deg)
(c)
180
20
0
–20
–40
0
150
40
–40
Fig. 2 Reconstructed (a) amplitude and (b)
phase of virtual images of the quarter wave
plate with the azimuth of  = 0° for the
horizontal polarization and for the vertical
polarization, respectively.
120
(b)
(a)
60
90
Rotation angle  (deg)
Rotation angle  (deg)
(b)
()
30
Ellipticity (deg)
A
A
100
180
0
: Theoretical
: Phase corrected
: Amplitude and phase corrected
30
60
90
120
150
180
Rotation angle  (deg)
(d)
Fig. 3 Theoretical and experimental values of polarization
parameters of object beams transmitted through QWP with various
azimuth angles : (a) amplitude ratio angle , (b) phase difference
, (c) principal axis angle , (d) ellipticity angle .
The reconstructed amplitude and phase of virtual images are shown in Fig. 2. Before reconstruction of the
holograms, the hologram intensity was Fourier transformed and only the Fourier spectrum component for the
virtual image in spatial frequency domain was filtered to remove a zero-order and conjugate image component,
and subjected to inverse Fourier transformation. The tilt of phase introduced by the off-axis geometry was
compensated for by adjusting the off-axis angle of the digital reference wave as described above.
The area A is for the reference (pol4) and B is for analysis (QWP) as seen in Fig. 2(a). The phase difference in
area A was averaged over a solid square area shown in the images (b) and the mean value was used to
compensate for the correct phase difference values in the area B. The mean value in the dashed square in the area
3
B was used for analysis. The derived polarization parameters are plotted by solid circle accompanied with their
theoretical values in Fig. 3. The theoretical polarization state of the object beam Uc is given by calculating product
between the Jones matrix of QWP and the Jones vector of linearly polarized incident beam as
 cos   sin   exp  i 4 

 sin  cos    0
Uc  
  cos  sin  1

  
exp  i 4     sin  cos  1
0
1  i  cos 2  sin 2 

.
2 1  i  cos 2  sin 2  
1
(1)
The theoretical values of , , , and  were derived by Eq. (1) and plotted as solid lines in Fig. 3.
The polarization parameters obtained by phase correction are in good agreement with the theoretical values.
The standard deviation of 3.09° is achieved for in the discrepancies between theoretical and experimental
values as seen in Table1In Fig. 3, some discrepancies of more than 10° between theoretical and experimental
-values can be found for  = 0°, 90°, and 180° because the polarization state is circular there.
The technique was also applied to observe a photoelastic effect induced in a PMMA plate of ring shape
compressed by a vise as shown in Fig. 4. The sample was 14.0 mm and 6.0 mm in outer and inner diameters,
respectively, and 3.0 mm in thickness. The area within the dashed rectangle in Fig. 4 was analyzed. The sample
was placed at the distance of 156 mm from the CCD surface. The incident beam was right circularly polarized by
inserting a QWP into the object beam after Pol3 in Fig. 1 and the hole of the sample acted as the reference area
described above. The polarizer Pol4 in Fig. 1 was removed in the experiment so that the reference was right
circularly polarized. Hence, the phase difference was supposed to be 90° in the reference area.
14.0 mm
6.0 mm
Force
Force
Vise
Vise
Hole
PMMA plate
Area for analysis
Fig. 4 Observation of photoelastic effect occurred in a
compressed PMMA ring. The ring was 14.0 mm in
diameter, 3.0 mm in thickness, which has a hole of
 at its center part. The hole of ring serves as a
reference area.
Figure 5 shows the reconstructed amplitude and phase
images and derived polarization parameters for (a) the
initial and (b) the stressed states, respectively. The gap of
the vise was equal to 14.0 mm and 13.6 mm for the initial
and stressed states, respectively. The area A in the
reconstructed image shows the reference area (the hole of
the ring) and B shows the analyzed area. The tilt of phase
in the area B of the reconstructed phase images was
eliminated by adjusting the digital reference wave so that
the tilted distribution of phase appeared in the area A. The
tilt effect appeared in the area A is cancelled in the
distributions of (X,Y) In the phase images after stress
application, the phase jumps between - and  appears. In
the distribution of (X,Y), the pattern of (b) the stressed
state considerably varies from that of (a) the initial state.
The difference in the distributions of polarization
parameters between (a) the initial and (b) the stressed
states shows a stress-induced birefringence of the sample.
4
B
A
(a)
(b)

A
A


 

A


 


A


3 mm
Fig. 5. Reconstructed amplitude (A⊥, A∥) and phase (⊥ ∥) of virtual image and analyzed
distributions of polarization parameters (, ) for (a) initial, (b) stressed states of the
ring, respectively.
5. Conclusions
We have developed a new digital holographic system for the polarization analysis by use of the fiber Faraday
rotator. By polarization switching of a single reference beam in an off-axis set up, the whole arrangement
becomes a compact and simple in analysis. In analysis compared with the previously reported method using two
reference beams, it does not require any additional positional matching of the reconstructed images because the
two orthogonal polarization reference beams take the same optical path. The reference area with known
polarization in the object is used for compensating for the phase drift of the reference beam during polarization
switching. Thus, it is possible to analyze successfully correct polarization state. It is shown that the proposed
technique can be applied to the photoelastic analysis of a stressed sample. The distribution of magnitude and
orientation of principal stresses may be obtained when the values of photoelastic constant, Young’s modulus,
Poisson ratio, and refractive index of the sample are known.
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Possible collaborations with other research areas
Digital holography can be applied for various measurements such as surface shape/deformation
measurements, polarimetry, photoelastic analysis, monitoring of drying process, etc. by means of
numerical amplitude and phase information of object wave [8-11]. In addition to holography, a
photometry is also one of our key researches. Optical metrology such as holography and photometry
widely covers the requirements of industrial and agricultural applications. To expand the applications of
the optical metrology, collaborations with other research areas are highly desirable.
(1) Collaboration with the high speed image processing for real time monitoring [8-14]
Noncontacting monitoring of the profile, deformation, drying process of surfaces, and plasma process that must
not be touched is often required in industry (processing industry, material testing, automobile industry…) and
daily lives (painting, nail polish,…). For such applications, digital holographic or interferometric techniques
with a high speed image processing are highly desirable.
(2) Collaboration with the 3D structural analysis [2, 3]
As shown in the manuscript, the digital holography can be applied for the measurement of birefringence, inner
stress. Especially, digital holography with a low coherence light sources is suitable for inner stress analysis. For
the purpose, collaboration with the 3D structural analysis is highly desirable.
(3) Collaboration with the certification techniques (information technology) [15]
Digital holography is also applied for information security techniques. The approach of watermarking in
digital holography has been explored. Collaboration with some information technology progresses the
application.
(4) Collaboration with the chemical engineering, environmental monitoring, electric circuit design to
develop a simple photometric sensor of practical use [16]
One of interesting applications of photometry is to develop a simple and practical optical sensor with LED light
sources for color detection of chemical reagents. Because of a very wide wavelength range of LED, it is very
easy to select a suitable wavelength for absorption band of target specimen. For example, the color changes of
chemical reagents for soil nutrients can be detected by means of a simple LED sensor. This kind of simple optical
sensor has a great impact on industrial and agricultural applications because of its compactness, simplicity, low
cost. To develop such kind of a simple photometric optical sensor, collaboration with environmental assessment,
chemical engineering, and electric circuit designer is desirable.
6
References
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pp. 1800-1801 (2003)
2. M. Yokota, Y. Terui, and I. Yamaguchi, “Whole-field analysis of polarization state by digital holography with
polarization modulation,” Opt. Rev., Vol. 13, No. 6, pp. 405-409 (2006)
3. M. Yokota, Y. Terui, and I. Yamaguchi, “Polarization analysis with digital holography by use of polarization
modulation for single reference beam,” Opt. Eng., to be published (2007)
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Materials Evaluation, Vol. 64, No. 7, pp. 724-730 (2006)
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with a wavelength shift,” Appl. Opt., Vol. 45, No. 29, pp. 7610-7616 (2006)
10. I. Yamaguchi, T. Ida, and M. Yokota, “Measurement of surface shape and position by phase-shifting digital holography
using dual wavelengths,” Journal of Holography and Speckle, Vol. 3, No. 2, pp. 98-105 (2006)
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Opt. Rev., to be published (2007)
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modulation of laser diodes,” Appl. Opt., Vol. 40, No. 28, pp.5023-5027 (2001)
13. M. Yokota, A. Asaka, and T. Yoshino, “Stabilization improvements of laser-diode closed-loop heterodyne
phase-shifting interferometer for surface profile measurement,” Appl. Opt., Vol. 42, No. 10, pp. 1805-1808 (2003)
14. F. Zhang, J. D. R. Valera, I. Yamaguchi, M. Yokota, and G. Mills, “Vibration analysis by phase shifting digital
holography,” Opt. Rev, Vol. 11, No. 5, pp. 297-299 (2004)
15. I. Yamaguchi, K. Yamamoto, G. Mills, and M. Yokota, “Image reconstruction only by phase in phase-shifting digital
holography,” Appl. Opt., Vol. 45, No. 5, pp. 975-983 (2006)
16. M. Yokota, T. Okada, and I. Yamaguchi, “Optical sensor for soil nutrient analysis by using LED light sources,” Meas.
Sci. Technol., to be published (2007)
Masayuki Yokota was born in Gunma, Japan, on January 31, 1967. He received the B.S.,
M.S., and Ph. D. degrees in electronic engineering from Gunma University in 1989, 1991,
and 1998, respectively. Currently, he is a research associate at the Department of Electronic
Engineering, Gunma University, where he has been working on optical engineering.
Dr. Yokota is a Member of the Japan Society of Applied Physics (JSAP), the Optical
Society of Japan (OSJ), the International Society of Optical Engineering (SPIE) and the
Optical Society of America (OSA).
7