Wave-plate phase shifting method
Zhigang Zhang
Qingchuan Zhang
Teng Cheng
Jie Gao
Xiaoping Wu
Downloaded From: http://spiedigitallibrary.org/ on 12/07/2013 Terms of Use: http://spiedl.org/terms
Optical Engineering 52(10), 103109 (October 2013)
Wave-plate phase shifting method
Zhigang Zhang
Qingchuan Zhang
Teng Cheng
Jie Gao
Xiaoping Wu
University of Science and Technology of China
Department of Modern Mechanics
CAS Key Laboratory of Mechanical Behavior and
Design of Materials
Hefei 230027, China
E-mail: [email protected]
Abstract. This paper proposes a new phase shifting method: wave-plate
phase shifting method. By different combinations of a quarter-wave-plate,
a half-wave-plate, and an analyzer, phase delays are introduced in the
interference light path in order to achieve the phase shifting digital holography. Theoretical analysis, numerical simulation, and experiments are
conducted to verify the validity of this method. The numerical simulation
shows that the result of the wave-plate phase shifting method is consistent
with that of the traditional four-step phase shifting method. The experimental results successfully reconstruct the object light intensity in the image
plane. Based on the wave-plate phase shifting method, a pixelated waveplate array structure is designed to achieve real-time phase shifting digital
holography. The wave-plate array phase shifting method not only can
reconstruct object image of high quality, but also can be used in dynamic
phase measurement. Therefore, pixelated wave-plate array structure and
wave-plate array phase shifting method could be widely used in practical
applications. © 2013 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10
.1117/1.OE.52.10.103109]
Subject terms: wave-plate phase shifting; digital holography; wave-plate array.
Paper 130949 received Jun. 27, 2013; revised manuscript received Sep. 21, 2013;
accepted for publication Sep. 24, 2013; published online Oct. 14, 2013.
1 Introduction
Phase shifting method is widely used to measure the phase in
optical interferometry. The basic idea of the phase shifting
method is that phase-shifts are introduced in the interference
light path to generate a controllable phase difference between
reference light and object light. Being calculated by multistep phase shifting, the object light phase is obtained to
improve the measurement accuracy. As for one specimen,
interferograms are acquired in different phase delays, and
then the whole phase distribution is obtained by a simple
calculation. Phase shifting method has been used in measurement,1–3 microscopic imaging,4–7 particle size analysis,8
three-dimensional display,9–11 image encryption,12,13 and
so on.
The traditional phase shifting method has several types
such as the time phase shifting (TPS) method and the spatial
phase shifting (SPS) method. The TPS method includes
the piezoelectric ceramic method,14 grating phase shifting
method,15 the tensile fiber method,16 the polarization
phase shifting method,17 and so on. The interferograms are
recorded at different times in the TPS method; thus this
method only can be used to measure the phase of a static or
quasi-static object. In the SPS method, the interferograms are
recorded at the same time; thus it can be used in dynamic
phase measurement. The SPS method mainly includes ordinary beam splitter þ polarization phase shifting method,18
grating splitter þ polarization phase shifting method,19 and
grating splitter þ grating phase shifting method.20 The SPS
method can be used in dynamic phase measurements; however, as the interferograms are recorded in different spatial
positions, the photoelectric performances of different detectors (if the interferograms are recorded by different detectors)
or different regions of the detector (if the interferograms
0091-3286/2013/$25.00 © 2013 SPIE
Optical Engineering
are recorded by different regions of a single detector) are
consistent, and the interferograms are required to be pixellevel position matched and gray-scale corrected. Thus, it is
difficult for the actual operation.
This paper proposes a wave-plate phase shifting (WPS)
method. In the second part, the light path schematic and
detailed theoretical derivation of the WPS method are given.
In the third part, numerical simulation is given to verify the
validity of the WPS method. In the fourth part, experiments
of observing a leaf stoma are conducted to verify the validity
of the WPS method. In the fifth part, based on the WPS
method, a pixelated wave-plate array (PWA) structure is presented and designed, and a real-time phase shifting digital
holography based on this structure is proposed.
2 Theory
The light path of the WPS method is shown in Fig. 1. The
light emitted by the laser illuminates a polarization beam
splitter after being expanded and collimated, and two
beams transmitted and reflected by the polarization beam
splitter are mutually perpendicular linearly polarized beams,
one of which is the object light that is reflected to the measured object, and the other is reference light that is transmitted. After being reflected by mirrors, the object light and the
reference light reach a second polarization beam splitter,
respectively, and interferograms are recorded by CCD after
the light transmitting wave-plates and an analyzer. Before
transmitting the wave-plates, the amplitude, frequency, and
initial phase of the object light are Es , ω, and φ1 , respectively; the amplitude, frequency, and initial phase of the
reference light are Er , ω, and φ2 , respectively. The polarization direction of the analyzer in front of the CCD is consistent with the polarization direction of the object light before
transmitting the wave-plates. The fast and slow axes of
the quarter-wave-plate (QWP) and half-wave-plate
(HWP)
⇀
are the f-axis (unit vector is represented by f ) and s-axis
103109-1
Downloaded From: http://spiedigitallibrary.org/ on 12/07/2013 Terms of Use: http://spiedl.org/terms
October 2013/Vol. 52(10)
Zhang et al.: Wave-plate phase shifting method
I 0 ¼ E2s ¼ I s :
(7)
2. When only the QWP is placed in the light path, the
phase delay of the slow axis with respect to the fast
axis is π∕2 (α ¼ π∕2). Then the amplitude and the
light intensity in the recording plane are shown in
Eqs. (8) and (9), respectively, and the derivation is
given in detail in Appendix A.
Eπ∕2 ¼ Es cos2 θ cosðωt þ φ1 Þ
þ Es sin2 θ cosðωt þ φ1 − π∕2Þ
þ Er sin θ cos θ cosðωt þ φ2 Þ
þ Er sin θ cos θ cosðωt þ φ2 þ π∕2Þ;
(8)
I π∕2 ¼ E2s ð1 − sin2 2θ∕2Þ þ E2r sin2 2θ∕2
Fig. 1 Light path schematic of the wave-plate phase shifting (WPS)
method. PBS, polarization beam splitter.
þ Es Er sin 4θ cosðΔφÞ∕2
þ Es Er sin 2θ sinðΔφÞ:
⇀
(unit vector is represented by s ), respectively. The angle
from the fast axis of the wave-plates to the polarization
direction of the analyzer is θ.
The object light and the reference light vectors are
expressed along the fast and slow axis directions of waveplates as
⇀
⇀
⇀
Es ¼ Es cos θ cosðωt þ φ1 Þf þ Es sin θ cosðωt þ φ1 Þ s ;
(1)
⇀
⇀
3. When only the HWP is placed in the light path, the
phase delay of the slow axis with respect to the fast
axis is πðα ¼ πÞ. Then the amplitude and the light
intensity in the recording plane are shown in Eqs. (10)
and (11), respectively.
Eπ ¼ Es ðcos2 θ − sin2 θÞ cosðωt þ φ1 Þ
þ2Er sin θ cos θ cosðωt þ φ2 Þ
¼ Es cos 2θ cosðωt þ φ1 Þ
⇀
Er ¼ Er sin θ cosðωt þ φ2 Þf − Er cos θ cosðωt þ φ2 Þ s :
(2)
After transmitting the wave-plates, the phase delay of the
slow axis with respect to the fast axis is α. Then
⇀
⇀
⇀
⇀
Er
⇀
⇀
þ Er sin 2θ cosðωt þ φ2 Þ;
The angle from the f-axis to the polarization direction of
the analyzer is θ; thus the interference light amplitude after
transmitting the analyzer is
þ Es Er sin 4θ cosðΔφÞ:
E3π∕2 ¼ Es cos2 θ cosðωt þ φ1 Þ þ Es sin2 θ cosðωt
2
þ φ1 − 3π∕2Þ þ Er sin θ cos θ cosðωt þ φ2 Þ
þ Er sin θ cos θ cosðωt þ φ2 Þ
− Er sin θ cos θ cosðωt þ φ2 − αÞ:
(11)
4. When both the QWP and the HWP are placed in the
light path, the phase delay of the slow axis with respect
to the fast axis is 3π∕2ðα ¼ 3π∕2Þ. Then the amplitude and the light intensity in the recording plane
are shown in Eqs. (12) and (13), respectively, and
the derivation is given in detail in Appendix B.
E ¼ Es cos θ cosðωt þ φ1 Þ þ Es sin θ cosðωt þ φ1 − αÞ
2
(10)
I π ¼ ðEs cos 2θÞ2 þ ðEr sin 2θÞ2
Es ¼ Es cos θ cosðωt þ φ1 Þf þ Es sin θ cosðωt þ φ1 − αÞ s ;
(3)
¼ Er sin θ cosðωt þ φ2 Þf − Er cos θ cosðωt þ φ2 − αÞ s :
(4)
(9)
− Er sin θ cos θ cosðωt þ φ2 − 3π∕2Þ;
(12)
(5)
I 3π∕2 ¼ E2s ð1 − sin2 2θ∕2Þ þ E2r sin2 2θ∕2
1. When neither the QWP nor the HWP is placed in the
light path, the phase delay of the slow axis with respect
to the fast axis is zero (α ¼ 0). Then the amplitude and
the light intensity in the recording plane are shown in
Eqs. (6) and (7), respectively.
E0 ¼ Es cosðωt þ φ1 Þ;
Optical Engineering
(6)
103109-2
Downloaded From: http://spiedigitallibrary.org/ on 12/07/2013 Terms of Use: http://spiedl.org/terms
þ Es Er sin 4θ cosðΔφÞ∕2
− Es Er sin 2θ sinðΔφÞ:
(13)
The above equations indicate that
I 0 þ I π ¼ I π∕2 þ I 3π∕2 :
(14)
October 2013/Vol. 52(10)
Zhang et al.: Wave-plate phase shifting method
The derivation is given in detail in Appendix C. Thus,
Eqs. (7), (9), (11), and (13) are not independent. Usually
all the four equations are used to improve the precision.
When θ is equal to 0 or π∕2, Eqs. (7), (9), (11), and (13)
only have one variable, Es . When θ is equal to π∕4 or 3π∕4,
then Eqs. (7), (9), (11), and (13) have variables Es , Er , and
sinðΔφÞ; thus the value of Δφ cannot be calculated.
Therefore, the value of θ could not be 0, π∕4, π∕2, or 3π∕4.
In order to simplify the calculation, π∕8 is chosen as the
value of θ. Then
I o ¼ E2s ;
(15)
I π∕2 ¼ 3E2s ∕4 þ E2r ∕4 þ Es Er cosðΔφÞ∕2
pffiffiffi
þ 2Es Er sinðΔφÞ∕2;
(16)
I π ¼ E2s ∕2 þ E2r ∕2 þ Es Er cosðΔφÞ;
(17)
I 3π∕2 ¼ 3E2s ∕4 þ E2r ∕4 þ Es Er cosðΔφÞ∕2
pffiffiffi
− 2Es Er sinðΔφÞ∕2:
(18)
It can be obtained from Eqs. (16) and (18) that
pffiffiffi
I π∕2 − I 3π∕2 ¼ 2Es Er sinðΔφÞ:
(19)
reference light is introduced and four frames of interferograms are generated in the recording plane while the phases
of the reference light are 0, π∕2, π, and 3π∕2, respectively.
Using these four interferograms, the intensity and phase of
the object light are reconstructed, see Figs. 2(c) and 2(d).
By the WPS method, four frames of interferograms are
acquired in the recording plane, see Figs. 2(i) to 2(l) (256×
256 pixel images captured in the center of the 1024 × 1024
pixel images); then the intensity and the phase of the object
light are reconstructed, see Figs. 2(e) and 2(f). The figures
show that intensity and phase reconstructed by the WPS
method is of high quality and basically the same as the intensity and phase reconstructed by the four-step phase shifting
method.
Mean square error (MSE) is used to quantitatively compare the reconstruction results. Variance is the mean value of
the square of difference between the original intensity and
the reconstructed intensity, and MSE is the square root of the
variance. The intensity image is 8 bit (0 to 255), and the
MSEs of four-step phase shifting method and WPS method
compared with the original intensity are 1.67 and 1.67,
respectively. As both the four-step phase shifting method
and the WPS method use four interferograms to reconstruct
the intensity images, the loss of data occurs only in changing
From Eqs. (17) and (19) and sin2 Δφ þ cos2 Δφ ¼ 1,
ðI π∕2 − I 3π∕2 Þ2 þ 2ðI π − I 0 ∕2 − I r ∕2Þ2 ¼ 2I 0 I r :
(20)
There is only one variable I r in Eq. (20); thus the value of
I r can be solved. That is, the value of Er can be solved. By
substituting the value of Er into Eqs. (17) and (19), the values
of cosðΔφÞ and sinðΔφÞ are solved; thus the value of Δφ can
be solved.
Therefore, the values of Es , Er , and Δφ can be solved
by this method, and the complex amplitude [uðx; yÞ ¼
Es · expðiΔφÞ] in the record plane is obtained. The
complex amplitude U ðX; YÞ can be reconstructed by Fresnel
transformation.
Z þ∞ Z þ∞
1
uðx; yÞ
UðX; YÞ ¼
iλz −∞ −∞
ðX − xÞ2 þ ðY − yÞ2
× exp ik z þ
dxdy:
2z
(21)
3 Numerical Simulation
In order to verify the validity of theoretical analysis, numerical simulation is conducted as follows. The original intensity
and phase of the object light in the object plane are shown
in Figs. 2(a) and 2(b), respectively. The pixel number of
the CCD is 1024 × 1024, and the pitch of the pixels is
10 μm. The distance from the object plane to the record
plane is 10 cm, and the wavelength of the laser is 532 nm.
The complex amplitude in the record plane is simulated
by Fresnel transformation of the object plane complex
amplitude. By traditional four-step phase shifting method,
Optical Engineering
Fig. 2 256 × 256 pixel images. Object plane: (a) intensity distribution,
(b) phase distribution. Images reconstructed by four-step phase shifting method: (c) intensity distribution, (d) phase distribution. Images
reconstructed by WPS method: (e) intensity distribution, (f) phase distribution. Images reconstructed by pixelated wave-plate array phase
shifting method: (g) intensity distribution, (h) phase distribution. (i) to
(l) are the images of I 0 , I π∕2 , I π , I 3π∕2 acquired by the WPS method in
the record plane.
103109-3
Downloaded From: http://spiedigitallibrary.org/ on 12/07/2013 Terms of Use: http://spiedl.org/terms
October 2013/Vol. 52(10)
Zhang et al.: Wave-plate phase shifting method
the intensity to an integer, so the MSEs of these two methods
are small and similar.
4 Experiment
To verify the validity of the WPS method by experiments, a
leaf stoma specimen is observed by a 100× microscope
objective. The schematic of the experimental light path is
shown in Fig. 1; the leaf stoma specimen is placed on the
position of the object to be observed and is irradiated by
the object light. The 100 × microscope objective is placed on
the position “a” in Fig. 1. CCD is placed ∼180 mm from the
image plane of the microscope objective; thus the image is
defocused. The complex amplitude in the record plane is calculated by the WPS method, and then the complex amplitude
in the image plane is reconstructed by Fresnel transform.
In the experiments, first, neither the QWP nor the HWP is
placed in the light path; then a light intensity image I 0 is
recorded. Second, only the QWP is placed in the light path;
then a light intensity image I π∕2 is recorded. Third, only the
HWP is placed in the light path; then a light intensity image
I π is recorded. Fourth, both the QWP and the HWP are
placed in the light path; then a light intensity image I 3π∕2
is recorded. Then the intensity and phase are reconstructed
by these four interferograms. Figure 3 shows the micrograph
of the leaf stoma, the interferogram when the HWP is placed
in the light path, the reference light image, and the light
intensity images reconstructed in different distances z by
Fresnel transform. Reconstruction distance z is selected
from 140 to 220 mm, and the interval is 10 mm. When the
reconstruction distance is chosen as 180 mm, the texture of
the object light image is clearest. That is, the distance from
the image plane of the 100 × microscope objective to the
record plane is ∼180 mm.
Numerical simulation and experiments verify the validity
of the WPS method, and based on this method, a real-time
phase shifting method is proposed in Sec. 5.
5 PWA and Real-Time Phase Shifting Digital
Holography
In 2004, Awatsuji et al.21 proposed quasi-phase shifting digital holography by adding a phase retarder array in the reference light path. The unit of the retarder array is aligned to the
pixel of the image sensor one by one, and the phase delays of
each 2 × 2 units in the array are 0, π∕2, π, 3π∕2. Four kinds
of optical path differences are introduced in one interferogram, so this method achieves real-time phase shifting digital
holography. However, this method is based on the four-step
phase shifting method, and the phase delays are caused by
different thicknesses of the glass units, so the phase delay
array must be placed in the reference light path, which means
the alignment is cumbersome. Then, Awatsuji’s group22,23
proposed real-time phase shifting digital holography method
based on the pixelated polarizer array, and the pixelated
polarizer array is integrated in front of the CCD. This method
solves the problem of alignment in the experiment, and the
real-time phase shifting digital holography is achieved.
The current mainstreams of the pixelated polarizer array
include the polarizer array based on iodide-containing polyvinyl alcohol (PVA) film24 and the polarizer array based on
metal nanograting.25 The thickness of the polarizer array
based on the PVA film is ∼50 um, which is much larger
than the size of CCD pixel; thus the crosstalk of the pixels
Optical Engineering
Fig. 3 Micrograph of the leaf stoma, interferogram, reference light
image, and reconstructed images in different distances.
is a problem.24 The performance of the polarizer array
based on the metal nanograting is superior, but as the period
of metal nanograting is only ∼100 nm and there are four
polarization directions in each 2 × 2 unit, the yield is very
low and the fabricating cost is very high. Four difficulties
are shown below in the fabrication process: (1) Due to the
diffraction limit of the ultraviolet (UV) light, the nanowires
cannot be fabricated by standard photolithographic techniques,
so holographic lithography or interference lithography is
needed.25 (2) In order to get coherent UV light source, frequency multiplication of visible laser is required. (3) To obtain
four polarization directions in each 2 × 2 unit, quartic of holographic lithography or interference lithography are needed.
(4) As the period of the metal nanograting is only ∼100 nm,
it is difficult to directly transfer the nanograting structure from
the photoresist template to the metal layer by etching process.
The cost of the polarizer array is very expensive and the array
is difficult to obtain; thus finding a cheaper and available structure instead of the polarizer array is significant.
Based on the WPS method, the PWA is designed to
achieve real-time phase shifting digital holography. The
PWA structure shown in Fig. 4 should be fabricated. The
phase delays of the slow axis with respect to the fast axis
of each 2 × 2 wave-plate units are 0, π∕2, π, and 3π∕2, respectively. The pitch of the PWA units is the same as the CCD
103109-4
Downloaded From: http://spiedigitallibrary.org/ on 12/07/2013 Terms of Use: http://spiedl.org/terms
October 2013/Vol. 52(10)
Zhang et al.: Wave-plate phase shifting method
Fig. 4 Assembling schematic of the pixelated wave-plate array
(PWA), analyzer, and CCD.
pixel pitch. The PWA and an analyzer are integrated in front of
the CCD, and it is important that the alignment of the PWA
unit with the CCD pixel one by one should be extremely precise. The installation angle from the fast axis of the PWA to
the polarization direction of the analyzer is π∕8. Compared
with the phase delay array proposed by Awatsuji in 2004,21
the PWA is integrated in front of the CCD, so the cumbersome
alignment in the experiment is not needed and the experimental complexity is greatly reduced.
PWA is first proposed and there is no product yet. One
possible method to fabricate the array is etching birefringent
material, and different phase delays are obtained by different
etching depths. It is difficult to precisely control the depth in
etching technology; thus this paper gives a feasible method
to fabricate PWA, see Fig. 5. In step 1, a glass substrate of
high transmittance is double-side polished. In step 2, a true
zero-order QWP is adhered to the substrate with UV-sensitive adhesive; then the plate is etched to get the phase delays
of 0, 0, 0 and π∕2 in each 2 × 2 unit. In step 3, a second true
zero-order QWP is adhered to the first QWP with UV-sensitive adhesive; then the second plate is etched to get the
phase delays of 0, 0, π∕2 and π∕2 in each 2 × 2 unit. In
step 4, a third true zero-order QWP is adhered to the second
QWP with UV-sensitive adhesive; then the third plate is
etched to get the phase delays of 0, π∕2, π∕2 and π∕2 in
each 2 × 2 unit. Therefore, the phase delays of each 2 × 2
unit in the PWA are 0, π∕2, π and 3π∕2, respectively. If larger
birefringence difference material (such as calcite) is used, the
thickness of a true zero-order QWP can be less than one
micron, and the thickness of a PWA can be achieved of
just a few microns.
In the wave-plate array phase shifting method, the complex amplitude of the object light can be calculated by only
one frame of image. The light intensity values of each 2 × 2
unit are acquired by CCD when the phase delays are 0, π∕2,
π, and 3π∕2, respectively, see Fig. 6(a). The method to reconstruct the object light complex amplitude in the recording
plane is shown as follows:
Optical Engineering
Fig. 5 Fabrication process schematic of the PWA.
1. Four images of null values are generated by computer,
and their resolutions are same as the resolution of the
image recorded by CCD.
2. The values whose phase delays are 0 in the CCD
image are copied to the corresponding positions of
the first computer-generated null value image. The values whose phase delays are π∕2 in the CCD image are
copied to the corresponding positions of the second
computer-generated null value image. The values
whose phase delays are π in the CCD image are copied
to the corresponding positions of the third computergenerated null value image. The values whose phase
delays are 3π∕2 in the CCD image are copied to the
corresponding positions of the fourth computer-generated null value image, see Fig. 6(b).
3. The remainder null values in the four computer-generated images are filled by linear interpolation method,
and then the four complete images are obtained, that
is, I 0 , I π∕2 , I π , and I 3π∕2 , see Fig. 6(c).
4. The complex amplitude of the object light in the record plane can be calculated by the four images generated above.
Simulation results of wave-plate array phase shifting
method are shown in Figs. 2(g) and 2(h), representing the
object light intensity and phase in the object plane. This
method successfully reconstructs the light intensity and
phase information of the object light. As one interferogram
is used to reconstruct the intensity images, the loss of data
occurs not only in changing the intensity to integer, but also
in the linear interpolation method. The MSE of the reconstructed intensity is 9.02, and the image quality of the
wave-plate array phase shifting method is a little worse
than the WPS method, because loss of information is introduced in the linear interpolation. Only one image is required
103109-5
Downloaded From: http://spiedigitallibrary.org/ on 12/07/2013 Terms of Use: http://spiedl.org/terms
October 2013/Vol. 52(10)
Zhang et al.: Wave-plate phase shifting method
Appendix A: Derivation from Eq. (8) to Eq. (9)
I π∕2 ¼ ½Es cos2 θ cosð−φ1 Þ þ Es sin2 θ cosðπ∕2 − φ1 Þ
þ Er sin θ cos θ cosð−φ1 Þ
þ Er sin θ cos θ cosð−φ2 − π∕2Þ2
þ ½Es cos2 θ sinð−φ1 Þ þ Es sin2 θ sinðπ∕2 − φ1 Þ
þ Er sin θ cos θ sinð−φ2 Þ
þ Er sin θ cos θ sinð−φ2 − π∕2Þ2
¼ ½Es cos2 θ cos φ1 þ Es sin2 θ sin φ1
þ Er sin θ cos θ cos φ2 − Er sin θ cos θ sin φ2 2
þ ½Es cos2 θ sin φ1 − Es sin2 θ cos φ1
þ Er sin θ cos θ sin φ2 þ Er sin θ cos θ cos φ2 2
¼ E2s cos4 θ þ E2s sin4 θ þ 2E2r sin2 θ cos2 θ
þ 2Es Er sin θ cos θ cos φ1 cos φ2 ðcos2 θ − sin2 θÞ
− 2Es Er sin θ cos θ cos φ1 sin φ2
þ 2Es Er sin θ cos θ sin φ1 cos φ2
þ 2Es Er sin θ cos θ sin φ1 sin φ2 ðcos2 θ − sin2 θÞ
¼ E2s ðcos4 θ − sin4 θÞ þ 2E2r sin2 θ cos2 θ
þ Es Er sin 2θ cos 2θ cos Δφ
Fig. 6 Reconstruction method.
þ Es Er sin 2θ sin Δφ
to reconstruct the complex amplitude of the object light; thus
this method can be used in real-time phase shifting digital
holography, and the loss of information can be decreased
with the trend of the CCD number increasing.
6 Conclusion
This paper presents a new WPS method in digital holography. In order to verify the validity of the WPS method,
numerical simulation is conducted and the complex amplitude of the object light is reconstructed successfully. In addition, the experiments of observing a leaf stoma are conducted
to verify this method. Using this method, the complex amplitude of object light is reconstructed successfully. Based on
the WPS method, a pixelated wave-plate array structure is
proposed. Wave-plate array phase shifting method not only
has the advantage of time phase shifting method so that an
object light image of high quality can be reconstructed,
but also can achieve the real-time phase measurement. Thus,
pixelated wave-plate array phase shifting method will contribute to the measurement of dynamically moving threedimensional objects, such as those of interest in biological
physics, fluid physics, particle measurement, and so on, and
this method will be widely researched in the future.
Appendix B: Derivation from Eq. (12)
to Eq. (13)
I 3π∕2 ¼ ½Es cos2 θ cosð−φ1 Þ þ Es sin2 θ cosðπ∕2 − φ1 Þ
Acknowledgments
This work was supported by the State Key Development
Program for Basic Research of China (Grant No.
2011CB302105), the National Natural Science Foundation
of China (Grant Nos. 11072233, 11127201, and 11102201),
and Instrument Developing Project of the Chinese Academy
of Sciences (Grant No. YZ201265).
Optical Engineering
þ Er sin θ cos θ cosð−φ2 Þ
þ Er sin θ cos θ cosð−φ2 − π∕2Þ2
þ ½Es cos2 θ sinð−φ1 Þ þ Es sin2 θ sinð−π∕2 − φ1 Þ
þ Er sin θ cos θ sinð−φ2 Þ
þ Er sin θ cos θ sinð−φ2 − π∕2Þ2
¼ ½Es cos2 θ cos φ1 − Es sin2 θ sin φ1
þ Er sin θ cos θ cos φ2 þ Er sin θ cos θ sin φ2 2
þ ½Es cos2 θ sin φ1 þ Es sin2 θ cos φ1
þ Er sin θ cos θ sin φ2 þ Er sin θ cos θ cos φ2 2
¼ E2s cos4 θ þ E2s sin4 θ þ 2E2r sin2 θ cos2 θ
þ 2Es Er sin θ cos θ cos φ1 cos φ2 ðcos2 θ − sin2 θÞ
þ 2Es Er sin θ cos θ cos φ1 sin φ2
− 2Es Er sin θ cos θ sin φ1 cos φ2
þ 2Es Er sin θ cos θ sin φ1 sin φ2 ðcos2 θ − sin2 θÞ
¼ E2s ðcos4 θ − sin4 θÞ þ 2E2r sin2 θ cos2 θ
103109-6
Downloaded From: http://spiedigitallibrary.org/ on 12/07/2013 Terms of Use: http://spiedl.org/terms
þ Es Er sin 2θ cos 2θ cos Δφ − Es Er sin 2θ sin Δφ
October 2013/Vol. 52(10)
Zhang et al.: Wave-plate phase shifting method
Appendix C: Derivation of I0 Iπ Iπ∕2 I3π∕2
I 0 þ I π ¼ E2s þ ðEs cos 2θÞ2 þ ðEr sin 2θÞ2
þ Es Er sin 4θ cosðΔφÞ
¼
E2s
þ ð1 þ cos2 2θÞ2 þ E2r sin2 2θ
þ Es Er sin 4θ cosðΔφÞ
I π∕2 þ I 3π∕2 ¼ E2s ð1 − sin2 2θ∕2Þ þ E2r sin2 2θ∕2
þ Es Er sin 4θ cosðΔφÞ∕2 þ Es Er sin 2θ sinðΔφÞ
þ E2s ð1 − sin2 2θ∕2Þ þ E2r sin2 2θ∕2
þ Es Er sin 4θ cosðΔφÞ∕2 − Es Er sin 2θ sinðΔφÞ
¼ E2s ð2 − sin 2θÞ þ E2r sin2 2θ þ Es Er sin 4θ cosðΔφÞ
¼ E2s ð1 þ cos2 2θÞ þ E2r sin2 2θ þ Es Er sin 4θ cosðΔφÞ
¼ I0 þ Iπ
References
1. P. Ferraro et al., “Digital holographic microscope with automatic focus
tracking by detection sample displacement in real time,” Opt. Lett.
28(14), 1257–1259 (2003).
2. X. Sang et al., “Applications of digital holography to measurements and
optical characterization,” Opt. Eng. 50(9), 091311 (2011).
3. J. L. Valin et al., “Methodology for analysis of displacement using
digital holography,” Opt. Laser Eng. 43(1), 99–111 (2005).
4. G. Coppola et al., “Digital holography microscope as tool for microelectromechanical systems characterization and design,” J. Micro/Nanolith.
MEMS MOEMS. 4(1), 013012 (2005).
5. D. Dirksen et al., “Lensless Fourier holography for digital holographic
interferometry on biological samples,” Opt. Laser Eng. 36(3), 241–249
(2001).
6. M. Kim et al., “Plane wave illumination for correct phase analysis and
alternative phase unwrapping in dual-type (transmission and reflection)
three-dimensional digital holographic microscopy,” Opt. Eng. 49(5),
055801 (2010).
7. F. Yi et al., “Automated segmentation of multiple red blood cells with
digital holographic microscopy,” J. Biomed. Opt. 18(2), 026006 (2013).
8. G. Shen and R. Wei, “Digital holography particle image velocimetry for
the measurement of 3Dt-3c flows,” Opt. Laser Eng. 43(10), 1039–1055
(2005).
9. X. Sang et al., “Three-dimensional display based on the holographic
functional screen,” Opt. Eng. 50(9), 091303 (2011).
10. H. Zhang, Q. Tan, and G. Jin, “Holographic display system of a threedimensional image with distortion-free magnification and zero-order
elimination,” Opt. Eng. 51(7), 075801 (2012).
11. Z. M. A. Lum et al., “Increasing pixel count of holograms for threedimensional holographic display by optical scan-tiling,” Opt. Eng.
52(1), 015802 (2013).
Optical Engineering
12. X. F. Meng et al., “Two-step phase shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006).
13. X. Shen, C. Lin, and D. Kong, “Fresnel-transform holographic encryption based on angular multiplexing and random-amplitude mask,” Opt.
Eng. 51(6), 068201 (2012).
14. J. H. Bruning et al., “Digital wavefront measuring interferometer for
testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703
(1974).
15. W. H. Stevenson, “Optical frequency shifting by means of a rotating
diffraction grating,” Appl. Opt. 9(3), 649–652 (1970).
16. C. Joenathan and B. M. Khorana, “Phase-measuring fiber optic electronic speckle pattern interferometer-phase step calibration and phase
drift minimization,” Opt. Eng. 31(2), 315–321 (1992).
17. H. Z. Hu, “Polarization heterodyne interferometry using a simple rotating analyzer. 1: Theory and error analysis,” Appl. Opt. 22(13), 2052–
2056 (1983).
18. A. L. Weijers, H. van Brug, and H. J. Frankena, “Polarization phase
stepping with a Savart element,” Appl. Opt. 37(22), 5150–5155 (1998).
19. K. M. Qian, H. Miao, and X. P. Wu, “Real-time polarization phase shifting technique for dynamic deformation measurement,” Opt. Lasers Eng.
31(4), 289–295 (1999).
20. O. Y. Kwon, “Multichannel phase shifting interferometer,” Opt. Lett.
9(2), 59–61 (1984).
21. Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting
digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
22. T. Tahara et al., “Parallel phase-shifting digital holographic microscopy,” Opt. Express 1(2), 610–616 (2010).
23. T. Tahara et al., “Comparative evaluation of the image-reconstruction algorithms of single-shot phase-shifting digital holography,”
J. Electron. Imaging 21(1), 013021 (2012).
24. V. Gruev et al., “Fabrication of a dual-tier thin film micro-polarization
array,” Opt. Express 15(8), 4994–5007 (2007).
25. V. Gruev, “Fabrication of a dual-layer aluminum nanowires polarization
filter array,” Opt. Express 19(24), 24361–24369 (2011).
Zhigang Zhang is pursuing his PhD in solid mechanics at the
University of Science and Technology of China (USTC). His research
interests are related to real-time phase shifting digital holography,
infrared imaging, and optical tweezers.
Qingchuan Zhang is a professor in the CAS Key Laboratory of
Mechanical Behavior and Design of Materials, USTC. His research
interests in his group currently range from the Portevin–Le
Chatelier effect in metal alloy, optical readout infrared imaging to
micro-biosensor with optical measurement methods.
Teng Cheng is an associate professor in the CAS Key Laboratory of
Mechanical Behavior and Design of Materials, USTC. His research
interests include optical readout infrared imaging and digital speckle
correlation measurement.
Jie Gao is a post doctor in the CAS Key Laboratory of Mechanical
Behavior and Design of Materials, USTC. He researches on
MEMS design and fabrication.
Xiaoping Wu is a professor in the CAS Key Laboratory of Mechanical
Behavior and Design of Materials, USTC. Her research interest is the
application of optical measurement methods in the field of advanced
science.
103109-7
Downloaded From: http://spiedigitallibrary.org/ on 12/07/2013 Terms of Use: http://spiedl.org/terms
October 2013/Vol. 52(10)