Vol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1273

Video-rate quantitative phase analysis by a
DIC microscope using a polarization camera
SHUHEI SHIBATA,1,2,* WATARU TAKANO,1,2 NATHAN HAGEN,1,2 MASARU
MATSUDA,3 AND YUKITOSHI OTANI1,2
1

Department of Optical Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi, Japan
Center for Optical Research and Education, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi,
Japan
3
Center for Bioscience Research and Education, Utsunomiya University, 350 Mine-machi, Utsunomiya,Tochigi, Japan*shibata_s@opt.utsunomiya-u.ac.jp
2

Abstract: This paper describes how to take advantage of the replacement of an intensity
camera with a polarization camera in a standard differential interference contrast (DIC)
microscope. Using a polarization camera enables snapshot quantitative phase analysis so that
real-time imaging of living transparent tissues become possible. Using our method, wequantify the phase measurement accuracy using a phantom consisting of glass beads
embedded in lacquer. In order to demonstrate these advantages, we image the pumping heartand blood flow in a living medaka egg.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction
Differential interference contrast (DIC) microscopy [1] is a widely used method for observing
tissues or cells in vivo without the need for applying florescent dyes. DIC microscopy allows
analysis of detailed structures, high sensitivity in detecting phase information, and has astrong capability for optical sectioning [2]. Many researchers have proposed methods for thequantitative phase measurement of volumes using transmission DIC microscopy [3,4], andsurface profile measurement using reflection DIC microscopy [5–7]. Ishiwata et al. proposeda retardation-modulated DIC (RM-DIC) microscope for quantitative measurement of planar
microstructure at a given focal depth [8,9], and to measure 3D phase volume by opticalsectioning [2]. However, these quantitative reconstruction methods [2–9] can requiresignificant computation for each phase image because of the need to acquire multiple images
for use with the phase shift technique.
We show below that it is possible to obtain a quantitative phase image from a single raw
frame, so that real-time video is possible for a DIC microscope using a polarization camera –a camera for which a micropolarizer array has been attached to a detector array [10–12].
While Fabre et al. have previously shown how to make use of a photonic crystal polarization
camera [11] with a DIC microscope, they used scanning to collect multiple raw images inorder to obtain a single phase image [13]. Thus, they were unable to demonstrate video
imaging of dynamic phase objects. We demonstrate that it is possible to capture
measurements at the native frame rate of the polarization camera, and that throughdeconvolution with the DIC system’s inverse MTF, we can convert the raw intensity data intoquantitative phase images.
In Section 2 below, we introduce how to calculate a quantitative phase image from one
image captured by a polarization camera. In Section 3, we evaluate the proposed method forgetting accurate phase measurement using a sample of glass beads inside a lacquer medium.
In order to demonstrate the advantages of our method, we also show a video-rate phasemeasurement of rapid motion in a living medaka egg.

#352734
Journal © 2019https://doi.org/10.1364/BOE.10.001273
Received 7 Dec 2018; revised 21 Jan 2019; accepted 23 Jan 2019; published 19 Feb 2019

Vol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1274

2. Snapshott phase analy
ysis methodd
We start withh a simple one-dimensional modelmconsistiing of a pure pphase object immaged bythe DIC microscope, geneerating an inteensity distributtion I(x) at thhe detector pllane. The
microscope consists of a lig
ght source, illu
umination optiics, the phase object, imaginng optics,
and a detectorr array (Fig. 1).

Fiig. 1. Simple modeel of a transmissioon microscope.

Accordingg to Hopkins’ theoryt[14] andd the partial cooherence theorry of a microsccope [15],the image inteensity distributtion I(x) on the detector arrayy can be definedd as
I ( x) =  −∞ R ( f x , f x′)O( f x )O∗ ( f x′) exp {−2π i ( f x − f x′) x} df x df x′,∞(1)

hase object o(xx), and R(fx, f'x) is the compleex-valued
where O(fx) iss the Fourier trransform of phtransmission cross-coefficiecent (TCC) givenn by∞

R( f x , f x′) = −∞ Q(ξ ) p(ξ + f x ) p ∗ (ξ + f x′)d ξ ,(2)for Q(ξ) the intensity distriibution of the illuminating ssystem, and p((ξ) the compleex-valued
pupil functionn.
The DIC microscope utilizesulateral shearing inteerference betw
ween two orthhogonallypolarized beam
ms produced byb two Nomarsski prisms, in w
which the beam
ms pass throughh slightly
different areaas (shear distaance Δ) of a specimen.
s
Thee two beams are then brouught back
together by a second paiir of Nomarsski prisms thaat cancel thee shear. The resulting
m observed at thet image planne encodes thee spatial gradieent of the objeect phaseinterferogram
along the sheaar direction.
Figure 2 showssour prop
posed optical layoutlfor the DIC microscoope using a polarization
camera. The DIC
D microscop
pe is set to geneerate π/2 of rettardation alongg the shear axiss between
the two Nomaarski prisms beecause the miccrostructure phhase is emphasiized best at thiis setting.
The polarizattion camera caan detect four kinds of intennsities with eaach polarizer aazimuthal
angle (0°, 45°°, 90°, and −455°) by one imag
ge. The intensiities detected uusing the pixelss oriented
at 0° and 90° of the microp
polarizer array do not producce interference because thesee axes arealigned to thee shearing direcction. Thereforre we use onlyy the two intenssities detected using the
pixels oriented at 45° and −45°
4 and ignoree the 0° and 90°° data.

Vol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1275

Fig. 2. The opttical setup of the DIC
D microscope ussing a polarizationn camera.

The polariization elements are defined withw the follow
wing Jones mattrices:5 ) = 
P (45

 2

 0


M ( ±45 ) = 
 , MPA
0 
± 2


2 
 0

,
0 

2 

N1 (Δ,ξ
Δ )=
,exp {iπ (ξΔ +1) 2} eexp (−iπξΔ 2)


0
0



(3)

Δ )=
N 2 (Δ,ξ.
exp (−iπξΔ 2) exp (iπξΔ 2)


0
0
where, P(45°)) and MPA(θ) are Jones mattrices of the poolarizer and thhe micro polariizer arrayoriented at θ = 0°, 45°, 90°, and −45°, andd N(Δ, ξ) is Jonnes matrix of thhe Nomarski prrism.
The TCC R(fx,f'x) in the casecof a polarization cameraa is rewritten using Eqs. (2) aand (3) as∞
R( f x , f x′) = −∞ Q(ξ ) Ap (ξ + f x ) A∗ p∗ (ξ + f x′)d ξ ,(4)

A = MPA(±455 )N1 ( Δ, ξ ) N 2 ( Δ , ξ ) P (45 ).

(5)

where

Accordingg to Ishiwata’s theory [8,9], we will asssume that thee object phasee can beapproximatedd as a small phaase distribution
n atop a (potenntially large) sppatially-constaant phase.
Therefore, phhase object o(x)) can be appro
oximated by itss Taylor expannsion about thee constantphase value aso( x) = C exp {iφ ( x)} ≈ C exp(iφ0 ) {1 + iφ ( x) − φ ( x) 2 } ,(6)for transmissiion C and spatiially-constant phasep
φ0. The F
Fourier transfoorm of phase obbject o(x)can be expressed as{}

1
O ( f x ) = C exp(iφ0 ) δ ( f x ) + iΦ ( f x ) − Φ ( f x ) ⊗ Φ * ( f x ) .
2(7)

where δ(fx) is the Dirac deltaa function, andd Φ(fx) is the F
Fourier transforrm of phase disstribution
φ(x). Here ⊗ is the convolution operatoor. Assuming tthat the pupil is independennt of any
polarization, thet intensity diistribution I(x,θθ) is rewritten bby substitutingg Eqs. (4) and ((7) to Eq.
(1). Since the introduction ofo the polarizatiion calculus haas converted R(fx,f'x) from a sscalar to aVol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1276

vector, we also need to convert the complex conjugate in Eq. (1) to an adjoint operator (i.e.
conjugate transpose), producing
C∞ cos(π f Δ ) M ( f )Φ ( f ) ⊗Φ * ( f ) exp( −2π if x ) df I ( x , ±45 ) =  M (0) − −∞
x
x
x
x
x
x
2


∞ {1+sin(2π f Δ )}m ( f )Φ ( f )Φ * ( f ) df
+ −∞
x
x
x x
0 x

∞

± 2i −∞ sin(π f x Δ)M ( f x )Φ ( f x ) exp(−2π if x x)df x
∞
−∞

(a)

(b)

(8)

±  cos(2π f x Δ)md ( f x )Φ ( f x )Φ (− f x ) exp(−4π if x x) df x , (c)
*

where
∞

∞

M (0) = −∞ Q(ξ ) p (ξ ) p* (ξ )d ξ , M ( f x ) = −∞ Q(ξ ) p(ξ + f x ) p* (ξ )d ξ ,∞∞m0 ( f x ) = −∞ Q(ξ ) p (ξ + f x ) p* (ξ + f x )d ξ , md ( f x ) = −∞ Q(ξ ) p (ξ + f x ) p* (ξ − f x )d ξ .

(9)Here M(0), M(fx), m0(fx), and md(fx) describe the illumination profiles in terms of the two
sheared beams illuminating the sample. We see that the intensity distribution I(x,θ) has fiveterms comprising three components (a), (b), and (c). The background component (a) consistsof M(0) plus an integral that evaluates to a constant – together these represent the DC part ofthe illumination and object transmission – plus an integral that expresses the illumination
profile due to the pair of illumination beams sheared by separation Δ. The second component(b) is the primary item of interest – the object phase gradient given by taking the difference
between the two shifted versions of the object phase. The difference operation is due to the
Fourier transform of the sine function being a pair of Dirac delta functions of opposite sign.
Finally, the third component (c) is the transmitted intensity profile dependent on the squared
object phase.
If we approximate the object as having a weak phase variation, the convolved phase2Φ ( f x ) ⊗ Φ * ( f x ) , the absolute value Φ ( f x )Φ * ( f x ) = Φ ( f x ) and the squared object phaseΦ ( f x )Φ * (− f x ) all become approximately zero. As a result, the intensity images (Eq. (8))
captured by the polarization camera depend only on the DC part of background M(0) and theobject phase gradient component (b). Thus, in the weak phase variation approximation Eq. (8)becomes:C∞
I ( x, ±45 ) =  M (0) ± 2i −∞ sin(π f x Δ )M ( f x )Φ ( f x ) exp(−2π if x x) df x  . (10)2The object phase gradient (the differential phase distribution) can be obtained by
I ( x, +45 ) − I ( x, −45 )2i ∞= sin(π f x Δ)M ( f x )Φ ( f x ) exp(−2π if x x)df xI ( x, +45 ) + I ( x, −45 ) M (0) −∞(11)∞= 2i −∞ MTF ( f x )Φ ( f x ) exp(−2π if x x)df x ,where
MTF ( f x ) =sin(π f x Δ) M ( f x )
M (0)

(12)

represents the Modulation Transfer Function (MTF) of the DIC microscope.
The phase distribution φ(x) of the observed object can be obtained using the MTF(fx) of
the DIC microscope with

Vol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1277

φ ( x) = I ( x, +45 ) − I ( x, −45 ) 1 −1 
1ℑ ℑ  ,
2i
 MTF ( f x )  I ( x, +45 ) + I ( x, −45 ) (13)

nction and ℑ−1 () is the inveerse Fourier ttransformwhere ℑ() is the Fourier transform funfunction.
on camera, thee image phasee distribution is measured uusing two
By using the polarizatio
on camera by E
Eq. (13). The sspatial resolutiion of thedifferent pixeel positions on the polarizatio
measured phaase is increasedd (worsened) byb a factor of 2 in each of thee two spatial ddirections.
Therefore, the polarization camera can experienceealiaasing error duee to insufficiennt spatialo overcome this problem, Tyo et al. proposed a Fourier-domaain pixel
sampling. To
interpolation method [16]. ThisT
method canc reconstructt a spatially baand-limited polarization
ut instantaneous field of vieww (IFOV) error by filtering inn the frequencyy domain.
image withou
We apply thiss method to th
he raw captured
d image in ordder to extract, II(x, 45°) and II(x, −45°)
prior to apply
ying them in Eq.
E (13). In ord
der to simplify the descriptioon of Tyo’s meethod, we
switch from thet Jones to thee Stokes-Muelller calculus, annd expand fromm one-dimension (x) totwo-dimensio
ons (x, y) whille maintaining
g the same immage model. A
At the DIC miicroscope
image plane th
he Stokes paraameters are given by(14)s ( x, y ) = I ( x, y, 0 ) + I ( x, y,90 ) = I ( x, y , 45 ) + I ( x, y , −45 ),0

s1 ( x, y ) = I ( x, y, 0 ) − I ( x, y,90 ),(15)

s2 ( x, y ) = I ( x, y, 45 ) − I ( x, y, −45 ),(16)

mage captured
d by the polaarization cameera for a pixeel whose
where I(x, y, θ) is the ima angle θ. WhileW
we do noot make use oof the 0° and 990° pixel
micropolarizeer is oriented atorientations in
n the phase im
maging analysis, we keep theem in the disccussion here inn order tokeep the explaanation of the filteringfmetho
od as clear as ppossible.
In the freq
quency domainn, (ζ, η) are Fourier spatial frrequency axes correspondingg to (x, y).
We write the Fourier transfo
orms of Stokess parameters s0 (x, y), s1(x, y),, and s2(x, y) ass S0(ζ, η),g the discrete-space Fourier trtransform (DSF
FT) of the imaage I(x, y,S1(ζ, η), and S2(ζ, η). Usingθ), we can separatelysfilteer the intensitty component S0(ζ, η), the {S1(ζ, η) + S2(ζ, η)}component, anda the {S1(ζ, η)−S
η 2(ζ, η)} component such as in Fig. 3. T
The interpolateed imagesof s0(x, y), s1(x,( y) and s2(x, y) can be reco
onstructed by 4 arithmetic opperations and aan inverse
DSFT after ussing a low-passs or high-pass filter.

Fig. 3.
3 The flowchart forf the Fourier-do
omain pixel interppolation method oof the polarizationn
camerra image.

After the pixel
pinterpolattion calibrationn, Eq. (13) bec omesφ ( x, y ) = s ( x , y ) 1 −1 
1
ℑ 
ℑ 2
 .
2i
 MTF ( f x )  s0 ( x, y ) (17)

mplementation for interpolatting the polariization cameraa’s pixels
This Fourrier-domain improvides the ideal filteringg method fromm the standpoiint of samplinng theory. As a result,Vol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1278

polarization error
ecaused byy edges of objeect features (i..e. instantaneouus field of vieew errors)
and aliasing errorseare minim
mized.

3. Evaluatio
on of quantita
ative analysis
A DIC microsscope (Olympu
us IX70) using a 4D Technol ogy PolarCamm (V model) polarization
camera is sho
own in Fig. 4. The light fromm a halogen lam
mp is transmittted through a 650 ± 25
nm bandpass filter into the DIC unit. The polarization ccamera exhibitss pixel extinctiion ratios
he pixel size iss 7.5 × 7.5 μm
m, the overall nnumber ofof ~30 acrosss the visible spectral range, thpixels is 648 × 460, and the maximum cammera frame ratee is 135 Hz.

Fig. 4. The DIC micro
oscope using a polaarization camera.

In our anaalysis, we use MTF(fx) calcu
ulated by Eq. (12) using thee illumination aperture,
wavelength, shears
distance Δ, NA and maagnification off the objective lens, then we calculate
1/MTF(fx) usiing a Wiener filter. Figure 5 shows the innverse MTF foor our system, showingthat the systeem suppressess low and meedium spatial frequencies (ffx < 0.4μm−1 and fx >0.6μm−1, for thet 20 × lens) whilew
maintaining the spatial frequencies neear 0.5 μm−1.

Fig. 5.
5 The reciprocal MTF(fx) of our DIC microscope. The red curves are for our 30 ×
objecttive lens (NA = 0.40), while the blacck curves are for oour 20 × objective lens (NA = 0.40).
The solid curve is the MTF
M at maximum
m aperture, while th
the dashed curve iis the MTF at halffure.
apertuIn order too test our apprroach quantitattively, we meaasure the phasee profile of glaass beads(refractive inddex n1 = 1.56)) embedded inn a lacquer meedium (refractiive index n2 = 1.54) toVol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1279

determine how well the system can estimate quantitative phase using the proposed analysis.
The diameter d of the glass beads is 2 μm. Figure 6 shows the measurement results of thedifferential phase on a region of our sample before the inverse Fourier transform (calculated
by Eq. (11), shown on the left side of the figure) and the quantitative phase after inverse
Fourier transform (calculated by Eq. (14), shown on the right side of the figure). The image
shown in the figure is sampled at 0.25 μm / pixel
We can see the well-known halo artifact indicated by negative phase values on both sidesof the glass beads. The theoretical curve is calculated from the difference of refractive index(n1 − n2 = 0.02), the bead diameter (d = 2 μm), and the wavelength (650 nm).
The halo artifact visible in Fig. 6(f) is caused by insufficient spatial coherence of theillumination system. Nguyen et al. have shown that they were able to successfully remove theartifact by using an external interferometric unit [17], but our system currently does not havea similar hardware setup allowing for removal of the halo. We obtain an absolute maximum
phase difference of 0.05 rad between our measurement curve and theoretical curve at theglass bead area without the halo artifact.

Fig. 6. The measurement results of the glass beads (n1 = 1.56) embedded in lacquer (n2 = 1.54).
We use an NA = 0.4 objective lens, (a), (c) and (e) are the differential phase results, while (b),(d), and (f) are the quantitative phase results. (c) and (d) are the enlarged images of red squareof (a) and (b), respectively. (e) and (f) are cross-sections taken at the center of the glass beads(red dished lines of (c) and (d)). The black line of (f) is the calculated theoretical value.

4. Video-rate quantitative analysis result

In biology, zebrafish and medaka are often used for investigating the tissue formation process
and for the observation of abnormal and healthy cells. Therefore, in order to demonstrate
video-rate quantitative phase measurement, we measure a living medaka egg (5~6 days after
spawning) (see Fig. 7). Figure 8 shows a 20 Hz video measurement of a medaka heart’s

Vol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1280

pumping and blood flow [188]. In this meassurement, the sspatial samplinng is 0.25 μm/ppixel. The
conventional method of quaantitative phasee measurementt (not video-ratte measuremennt) cannot
take detailed phase imagees because off the fast mootion, while oour method (vvideo-rate
measurement)) using the po
olarization cam
mera can obtainn clear phase images of thee working
valve betweenn the atrium annd the ventriclee, and movemeent of red bloood cells even dduring theheart’s pumpiing action.

Fig. 7. A living medaka egg
g as sample for viddeo-rate measurem
ment.

Fig. 8.
8 Two frames frrom a quantitativee phase video off the pumping off a medaka heart,
capturred at a 20 Hz framme rate [18]. (Visuualization 1).

5. Conclusio
on
Adding a polarization cam
mera to a DIC
C microscope allows video-rrate quantitatiive phasemeasurement of transparentt samples. We can extract thhe differential phase componnent fromges captured by the polarizattion camera, aand reconstructt the quantitatiive phase
the DIC imag
distribution frrom the differeential phase by our proposed aanalysis. Our mmethod can meeasure thedepth-integratted phase to ann accuracy of betterbthan 0.055 rad. Conventiional quantitattive phasemicroscopy withoutw
using a polarization camera cannoot measure dynnamic phase oobjects orobject that movemquickly duringdmeasureement. Our meethod, howeverr, can achievee accurate
measurement of these dynaamic objects deespite their moovement. As a result, we cann observes
the structures of the living tiissue without staining.

Funding
MEXT.

Acknowledg
gements
We would like to thank Prof. Toyohiiko Yatagai, D
Dr. Wataru IIshiwata and Olympus
f useful discu
ussions, for pro
oviding the DIIC microscopee, and for the D
DIC MTF
Corporation for
information foor this work. ThisT work was supportedsby thhe MEXT Optto-Bio Project.

Vol. 10, No. 3 | 1 Mar 2019 | BIOMEDICAL OPTICS EXPRESS 1281

Disclosures

The authors declare that there are no conflicts of interest related to this article.
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Vol. 10, No. 6 | 1 Jun 2019 | BIOMEDICAL OPTICS EXPRESS 2967

Video-rate quantitative phase analysis by a DIC
microscope using a polarization camera: errata
N ATHAN H AGEN , 1,2,* S HUHEI S HIBATA , 1,2 WATARU TAKANO, 1,2
M ASARU M ATSUDA , 3 AND Y UKITOSHI OTANI 1,2
1 Department of Optical Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi, Japan
2 Center for Optical Research and Education, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi,

Japan
3 Center for Bioscience Research and Education, Utsunomiya University, 350 Mine-machi, Utsunomiya,Tochigi, Japan* nh@hagenlab.org

Abstract: We correct several errors in the derivation of the MTF shown in our original
manuscript.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

In our original manuscript, [1] Eqs (1) through (9) provided a derivation of the MTF for apartial-coherence measurement model of a DIC microscope. Our first error was a typo appearing
in Eq. (6), where one of the terms is missing a factor of 12 . The correct Eq. (6) is:o(x) = C exp{iφ(x)} ≈ C exp(iφ0 ) 1 + iφ(x) − 12 φ2 (x)Our second error was in assuming that the polarization matrices that we originally obtained
from Ref. 2 were correct. On reflection, it is obvious that they cannot be. Ref. 2 gives the Jones
matrices for a 45◦ linear polarizer as√© 2 0ªP(45 ) = ­
√ ®,
2¬
«0
◦

which represents an ND filter rather than a polarizer. A valid Jones matrix for a polarizer would
instead be1 ©1 1ªP(45◦ ) = ­
®.
2 1 1
«
¬
Likewise, the matrices representing the retardances induced by the Nomarski prisms, N1 and N2 ,were given by Ishiwata et al. as©e
N1 = ­«iπ(ξ∆+θ)/20e−iπξ∆/2 ª®
0 ¬−iπξ∆/2and©e
N2 = ­« 0eiπξ∆/2 ª®.
0 ¬These are not a valid retarder matrices, and should instead be©1
N1 = ­«00ª®
iπ(ξ∆+θ)/2
e
¬

and

©1
N2 = ­«00e−iπξ∆/2ª®.
¬

These when combined together form the matrix A = P(±45◦ )N2 N1 P(45◦ ) for the two analyzed
polarization directions.
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Journal © 2019https://doi.org/10.1364/BOE.10.002967
Received 10 May 2019; published 23 May 2019

Vol. 10, No. 6 | 1 Jun 2019 | BIOMEDICAL OPTICS EXPRESS 2968

When these revised forms for the polarization matrices are inserted into the equation for thetransmission cross-coefficient R, we obtain∫ ∞R( fx, fx0) =A p(ξ + fx )A∗ p∗ (ξ + fx0) Q(ξ) dξ ,−∞so that R must take the form of a 2 × 1 vector rather than the scalar represented in Ref. 2 and ouroriginal manuscript. Thus, the illumination model Q(ξ) too must be expanded to take the formof a 2 × 1 polarization vector rather than a scalar. Finally, the equation representing the intensity
distribution at the detector array, Eq. (1) in our original manuscript, should be calculated as thesum of squares of the two elements of the polarization vector. Since A has the simple form
©1
A = 2(1 + cos θ) ­
«1

1ª
®,
1¬

i.e. a scalar factor times the uniform matrix, all of these changes turn out to have no effect on
the final result, Eqs (8) and (9) in the original manuscript. Indeed, the lack of effect on the
result is how we missed the original error. Thus, the derivation was faulty but the result remains
unchanged.
In addition, in the statement below Eq. (9), we stated that the background component (a) of
Eq. (8) consists of M(0) plus an integral that evaluates to a constant. Due to the presence of thevariable x inside the integral, however, it does not evaluate to a constant but instead to a functionof x.
The remainder of the paper, including the experimental results, is unaffected by each of these
corrections.
Finally, in the acknowledgements, Dr. Ishiwata’s first name was given incorrectly as Wataru,
but should have been given as Hiroshi.
References
1. S. Shibata, W. Takano, N. Hagen, M. Matsuda, and Y. Otani, “Video-rate quantitative phase analysis by a DIC
microscope using a polarization camera,” Biomed. Opt. Express 10, 1273–1281 (2019).
2. H. Ishiwata, M. Itoh, and T. Yatagai, “A new method of three-dimensional measurement by differential interference
contrast microscope,” Opt. Comm. 260, 117–126 (2006).