Interferometric Approaches to Evaluation of Strokes Camera and Object Rotation
2233005 Akter Monia
Principal Supervisor: Prof. Yoko Miyamoto
Supervisor:：Prof. Eriko Watanabe

1. Introduction

without introducing spatial shifts, limiting

There are two projects under the mentioned

the spatial resolution of such measurements.

title. One is an interferometric method toA dedicated Stokes camera can remove theevaluate the spatial resolution of Stokes

necessity for external analyzers. But Stokes

camera using polarization fringes withcameras are not necessarily designed todifferent spatial frequencies. Another is anrecord polarizations states that vary rapidlyinterferometric method for measuring thein space. In this work we evaluate a Stokes

angle of rotation in the object plane in terms

camera in terms of how well it can record theof the shift of the spectral peak.

polarization state that is changing rapidly in

2. Theory

space.

2.1 Stokes Parameters and Stokes Camera

2.2 Principle of Stokes Camera

The Stokes parameter is a quantity thatIn Figure 2.1 we show a possible hardware

expresses the polarization state of light. The

set up for a Stokes camera with an array of

Stokes parameters are a set of values thatpolarization filters. Only the intensity of adescribe

certain polarization component among thethepolarization

stateofanelectromagnetic oscillation.
S0 = IVertical + IHorizontal

horizontally,vertically,(45°)

rightandright

diagonallycircularly

polarized

components can pass through the filter.

=IRight Diagonal + ILeft Diagonal= IRight Circular + ILeft Circular ;S1 = IHorizontal − IVertical ;Figure 2.1: Polarization filter array andS2 = IRight Diagonal − ILeft Diagonal ;virtual super pixel made of real pixels of
S3 = IRight Circular − ILeft Circular (2.1.1)

Stokes camera.

Stokes camera measures the polarization

If we consider a virtual super pixel made ofstates of an optical field as a function offour real pixels, then the number of virtual

position by combining intensity distributions

super pixels will be one-fourth of the total

recordedpolarization

number of real pixels of the camera. Eachanalyzer settings. It is not always easy tovirtual super pixel can measure all the fourchange the polarization analyzer settings

polarization components in its four real

under

different1

pixels to calculate the S0 to S3 Stokes
parameters using equations (2.1.1).
3. Performance

evaluationofStokes

cameraIn this work we report an interferometricmethod to evaluate the spatial resolution of(a)Stokes cameras. The method is applied to theHWP = Half Wave Plate, QWP = Quarter Wave Plate,evaluation of a Photonic Lattice WPI200

BS = Beam Splitter, M = Mirror

camera. The interferometer is used to create
variationsofpolarization

stateswithdifferent spatial frequencies, and contrast inrecorded Stokes parameters is measured.
3.1 Experiment

(b)

(c)

The experimental setup is shown in Fig.

Figure 3.1: (a)Interferometer (b)3.1(a). A HeNe laser beam is expanded andInterference between diagonally polarized

collimated. The beam passes through either

light beams (c) Interference betweena polarizer (to create a diagonally polarizedcircularly polarized light beams.

beam) or a quarter wave plate (to create a4. Data analysis

circularly polarized beam), then enters anThe fringes are analyzed for signs ofinterferometer. Due to the number ofdegradation of contrast as the spatialreflections the beams passing through thefrequency increases.

two arms of the interferometer become

4.1 Calculation of Spatial Frequency

orthogonally polarized and create fringes in

Figure 4.1(a) is a schematic diagram of theStokes parameters S1 and S3 (interferenceinterference

between diagonally polarized beams), or S1

frequency is 1/L, L corresponds to theand S2 (interference between circularlyintervalpolarized beams), which is recorded by theinterference fringe.

fringe.

between

Whenthethepeaks

spatialoftheStokes camera. Fringe spacings can bechanged by adjusting Mirrors-1&2. The
Stokes camera used in the experiment iscontrolled by the dedicated software WPIViewer. The recorded spatial distribution of
S1 is shown in Figs. 3.1(b) and 3.1(c).

(a)

2calculated

from

thecross-section

ofinterference fringes as shown in figure 4.2.
Since the data points contain noise, thevisibility has been calculated by comparing
the data with an ideal sine wave instead ofusing the data points directly. Contrast has(b)been calculated as a ratio of S1 change andFigure 4.1: Theoretical model of calculating line spacing (L)and spatial frequency (f) from the recorded fringe pattern(a)and cross-section of the S1 Stokes parameter spatial
distribution (b).

average S1. S1 change has been denoted as X ′in equation and (average S1)+1 has been
denoted as Y’. Contrast C is the ratio of X ′and Y’ and can written as equation (4.2.1).

Cross-sections of the interference fringes
can be obtained in the x- and y-directions,
respectively, and there are different spatial
frequencies for each. The spatial frequency
in the x-direction is 1/Lx = fx , and that in
the y-direction is 1/Ly = fy . Fringe spacing
L can be calculated by using equation (4.1.1)
and spatial frequency f can be calculated

Figure 4.2: Comparison with ideal sinewave.

from the fringe spacing L.
12

12

L = 1/√( ) + ( )
Lx

and1

f=L

Ly(4.1.1)

C=X′Y′

S1max+1 −S1min+1= S1max+12+S1min+1(4.1.2)A cross-section of the S1 Stokes parameter

1 2

1 2X

Y

ΔC = |C|√( ′ ) + ( ′ ) Δ =spatial distribution is shown in figure 4.1(b).
The cross-section in figure 4.1(b) has been

2of a range of stable oscillations. Stableoscillation is an oscillation which has arelatively constant average and a relatively
constant amplitude.

4.2 Calculation of contrast
In this study, the visibility is calculated as ameasure of contrast. The amplitude is2

11|C|√( S1max+1−S1min+1 ) + ( S1max+1+S1min+1) ΔY2

compared with a sinusoidal wave with noise.
P1 and P2 are two points marking the ends

(4.2.1)22(4.2.2)
The range of C which is Δ C has been
calculated for all records. In this case,
Δ is defined as follows2⊿=√(ΔSmax ) + (ΔSmin )5. Assessing2(4.2.3)

2
Contrastwith

SpatialFrequencies

3

Contrast vs. spatial frequency for diagonally

6. Result:polarized interfering light beams is shown inFor

figures 5.1(a)and 5.1(b).

polarized

interference

betweenbeams,

diagonallycontrast

forStokes

parameter S1 (fig 5.1a) is independent of
spatial frequency in the range 0.12 to 0.27
pixel-1 . For S3 in figure 5.1(b), contrast isindependent of spatial frequency in therange 0.11 to 0.35 pixel-1. For interference(a)between circularly polarized beams, contrast
for Stokes parameter S1 in figure 5.2(a) isindependent of spatial frequency in therange 0.06 to 0.157 pixel-1. For S2 in figure
5.2(b), contrast is independent of spatial
frequency in the range 0.04 to 0.22 pixel-1.

(b)Contrast is low for interference between

Figure 5.1 Contrast vs. spatial frequency forcircularly polarized light beams.

(a) S1 and (b)S3 for interference between

7. Measurement

diagonally polarized beams.

Rotation
Luof

etIn-plane

Contrast vs. spatial frequency for circularly

Previously,

polarized interfering light beams has been

demonstrated

shown in figures 5.2(a) and 5.2(b).

measuring rotation in terms of spatial

aal.

Objectproposedtechniqueandbasedon

distribution of horizontal movement [1].
The group extracted this movement as phasedifference.

Theycalculated

thephase

gradient from two points that were close
together on the object surface to avoid phase(a)unwrapping, limiting the accuracy of themeasurement. Later Yang improved thetechniquebyexaminingfrequency

spectruminthespatialasimilarinterferometer, where the rotation angle has
determined from the shift of the spectral(b)peak [2]. However, the detection of the peakFigure-5.2: Contrast vs. spatial frequency

was done manually with limited resolution.

for (a) S1 and (b)S2 for interference

In this work I have replicated Yang’s results

between circularly polarized beams.

with a more compact interferometer.

47.1 TheoryΩ = arcsin [λ4πsinθThe general expression for the phase changeλ∙∂(∆φ)∂y] (7.5)

distribution ∆φ(x, y, z) is proportional toIn the above equation

the projection of the displacement vector ⃗lthat will be determined by the interferometer.

4π Sinθis a real number⃗⃗(u(x, y, z) onto the sensitivity vector k⃗⃗,v⃗⃗,w⃗⃗⃗⃗).
⃗⃗(u∆φ(x, y, z)=k⃗⃗,v⃗⃗,w⃗⃗⃗⃗) . ⃗l(x,y,z)(7.1)

7.2 Experiment

lx ,ly , and lz are the x, y, and z components

A He-Ne laser of wavelength 632.8nm isof the displacement vector. u⃗⃗⃗⃗ , v⃗⃗⃗ andexpanded. The collimating lens (f=100mm)⃗⃗⃗⃗⃗denote the unit vectors along the x, y, andw

lens is intentionally kept near the spatialz axis. The phase change caused by thefilter (35mm) to get a larger beam area onobject’s arbitrary rotation can be determined

the test object. The test object is anby

aluminum plate with rough surface.
∆φ(x, y, z) = φafter − φbefore =⃗⃗⃗⃗1 − ⃗⃗⃗⃗⃗⃗(kk 21 )l(x, y, z)⃗⃗⃗⃗1 − ⃗⃗⃗⃗⃗⃗−(kk 22 )l(x + ∆x, y, z) (7.2)

λ is the wavelength of laser, ⃗⃗⃗⃗
k1 denotes
the wave vector of the incidence beam, while⃗⃗⃗⃗⃗⃗
k 21 and ⃗⃗⃗⃗⃗⃗
k 22 are the wave vectors of twoobject beams; α and β are the incidence
angles.
2π⃗⃗⃗⃗k1 = − (w⃗⃗⃗ )λ⃗⃗⃗ cos α)w&⃗⃗⃗ cos β)
w

(7.3)

2π
⃗⃗⃗⃗⃗⃗k 21 = (u⃗ sin α +;λ2π⃗⃗⃗⃗⃗⃗
k 22 = (−u⃗ sin β +λ(7.2) can be written as2π
λ[2 sin θ ∙ lx + sin θ ∙∂lx

To make the imaging a little easier, a metal∂xof the beam. The incident beam getsscattered after hitting the test object. Two∙ ∆x](7.4)

scattered beams are selected by apertures(diameter of 1mm). The two beams are thenΩ is the rotation angle of the object and lx
is the displacement in x direction,lx = x ′ − x = xcosΩ-ysin Ω-x
Partial derivative of lx with respect to y andx areinterferogram (top right).

wire is pasted on the test object at the center

We consider that θ = α = β . Then equation∆φ(x, y) = −Figure-7.4: Optical setup and recorded∂lx∂y= −sinΩ and∂lx∂x= cosΩ − 1 ,reflected by a prism towards the imaging lensand are observed on the image sensor of theCCD camera.

Interferograms are obtained

as the object is rotated in steps of 0.01°by
using a rotation stage.
7.1 Fourier transform approach of In-plane

respectively.
By putting the formula for(7.4), we can show that∂lx∂xobject rotation.
in equation

The intensity of the interferometric speckle
pattern before and after rotation of the test5

object can be written as equation 7.1.1 andD(fx , fy ) = DΩ=0 (fx , fy − fs Ω)equation 7.1.2:

fs Ω is the amount of shift of spectral peak inIbefore (x, y) = a(x, y) + cbefore (x, y)ei2π(f0xx) +fy direction and has been observed as shown∗(x, y)e−i2π(f0xx) (7.1.1)
cbefore(7.1.8)in figure-7.5.

Iafter (x, y)=a(x, y)∗+ cafter (x,y)e2πi (f0xx) + cafter(x,y)e-2πi(f0xx) .

7.6 Result(7.1.2)

With the increase of the rotation angle, it has

We apply Fourier transform to the above

been observed that the spectral peak shifts inequations (7.1.1) and (7.1.2) and denote as

the y direction as in Yang’s work. The

Ibefore FT androtation angle is expected to be proportional

Iafter FT respectively. They

are given by

to the amount of shift of the spectral peak.

Ibefore FT (fx , fy ) = A(fx , fy ) + Cbefore (fx −Compared to Lu’s method, this technique is∗
f0x , fy ) + Cbefore(−fx − f0x , −fy )expected to have less error because it uses

(7.1.3)Iafter FT (fx , fy )=A(fx ,fy )+Cafter (fx −information from the entire image instead of∗
f0x , fy )+Cafter(−fx − f0x , −fy ).

two close points.

cbefore (x, y)andcafter (x,y)(7.1.4)canbereconstructed by filtering, shifting, andSummary

applying inverse transform to Cbefore andIn this paper, the performance evaluation ofCafter .

the spatial resolution of the WPI-200 Stokes

We definecamera has been done. Polarization fringes∗d(x, y) = cbefore (x, y) cafter(x, y)(7.1.5)and its Fourier transform
F[d(x,y) ] = D(fx , fy ).

of different spatial frequencies have beenrecorded. Spatial resolution has been found(7.1.6)

to be independent of the spatial frequency.
A compact optical system has been realized
for the measurement of in-plane object
rotation angle. The interferograms havebeen recorded

andpartially processed

according to the proposed method, and theFigure-7.5: (a) Phase distribution of∗
cbefore (0.00°) × cafter(0.03°) (b) Fourier

shift in frequency space has been observed.
References[1] Min Lu et al., Opt. Lett. 42, 1986 (2017).

transform of (a)For small rotation we can show that[2] YANG

cafter (x,y) ∼ cbefore (x, y)e-2πifsΩy , so that

面内回転角測定の実験的検証」 , 電気通信大学 修士論d(x, y) = dΩ=0 (x, y)e2πifsΩy ,

文 , (2022)(7.1.7)

ANDONG 「空間周波数成分に注目した

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