Method of Measuring the Rotation Angle Based on Uniaxial Crystal Conoscopic Interference in Polarimeter

Huang Yantang , Ou Ling Wang Jincheng Physics and Information Engineering College , Fuzhou University,Fu Zhou, P. R. China 350108

ABSTRACT

A method of measuring the rotation angle in polarimeter is proposed .To analysis uniaxial crystal convergence polarization-interference phenomenon with Jones Matrix, the polarizer rotates an angle relatively to the analyzer , the relationship between the rotation angle of cross-shape pattern θ1 and the polarized plane rotation angleθ

2caused by the

polarizer (or optical activity) is deduced:θ2 =2θ1.This rule can be used to measure the rotation angle in polarimeter: the

polarized plane will rotate when it transmits the optical activity substance ,so the angle of cross-shape pattern will rotate. Using CCD camera to shoot the conoscopic interference pattern, measure the angle of cross-shape pattern rotated by image processing technology to obtain the angle of polarized plane rotation. Experiment is designed to demonstrate this method. Comparing the method of the stepping-motor rotating the analyzer to detect the dark position and measure the angle, this new method is real-time , digital and automation for measuring the rotatory power in polarimeter. Keywords： Polarimeter, Optical Activity , Conoscopic Interference Pattern, Optical Rotation Angle , Cross-shape

1. INTRODUCTION

When a linearly polarized light wave is passed through some substances(such as sucrose solution), it is observed that the emerging wave has its E-vector (plane of polarization) rotated. The rotation angle is proportional to the medium thickness and the concentration the solution .The rotation of the plane of polarization by a substance is called optical activity. With measured the rotation angle of the plane of polarization ,the optical activity characteristics will determine, and deduce the concentration and ingredients .The polarimeters widely used in various fields of sugar, biochemical, food and organic chemical. The conventional method of measuring the rotation angle of the plane of polarization is extinction technique (Fig.1). Before measuring the sample , adjust the analyzer (P2) to be crossed the polarizer(P1) to view dark window, write down the degree scale of P2 with φ1 . When put in the sample , the light transmit P2 to emerge bright view window because of optical activity . To adjust the analyzer (P2) to view dark window again, write down the degree scale of P2 withφ2 . The rotation angle of P2 is Δφ=φ2-φ1, is called the rotatory angle of the linearly polarized light wave passed through the sample. Because of the limits of the human eye resolving power, this method is low accuracy .

Fig. 1 Schematic diagram of measuring the rotation angle of the polarized light

Over the past decade ，in order to realize the automatic measuring the rotatory angle , several researchers have proposed methods to improved polarimeter[1~2] .For example , LI Bin used photodetector to measure the optical power passing through the the analyzer P2 to search for the extinction position with stepper motor to drive the analyzer P2. The angle of the stepper motor indicated form the initial position to the extinction position is the rotatory angle [1]. There are also the shortages of mechanical error coming from stepping motor and can not adapt to real-time measuring the

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rotatory angle. Using CCD camera to shoot the conoscopic interference pattern, measured the angle of cross-shape pattern rotated by image processing technology to obtain the angle of polarized plane rotation.

2. MEASURING ROTATORY ANGLE WITH UNIAXIAL CRYSTAL CONOSCOPIC POLARIZED LIGHT INTERFERENCE METHOD

2.1 Conoscopic light interference The principle of a linearly polarized light wave interference is as follow: an incident beam become linearly polarized light after pass through the polarizer P1( Fig.2). After entering birefringent crystal ( such as LiNO3 crystal), the polarized light split into two orthogonally polarized waves, o-wave and e-wave with certain phase difference.When the two o-ray and e-ray pass through the analyzer, only the components along the polarization direction of the analyzer P2 can be passed.The components are coherence , producing conoscopic light interference pattern.[3~8]

Figure 2 Principle of a linearly polarized light wave interference

We setup coordinate system according to the axis of uniaxial crystal（Fig.3(a)）,and to the direction of the analyzer P2(Fig.3(b)), derivation the light intensity distribution formula of conoscopic polarized light interference pattern .

(a) (b)

(a) Setup coordinate system by axis of uniaxis crystal (e and o). P1，P2 is the direction of polarizer and analyzer (b) Setup coordinate

system by direction of polarizer. P2 ; P1，P2 is the direction of the polarizer and the analyzer. 1θ is the angle of x to e， 2θ is the angle x to P1

Figure 3 Conoscopic polarized light interference coordinates

As shown in Fig.3, the Jones vector of polarized light (a linearly polarized light wave passed through P1) in o-e coordinate system[9] is

2 1

2 1

cos( )sin( ) ie δ

θ θθ θ

−−

In x-y coordinates system is

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1 1 2 1

1 1 2 1

1 2 1 1 2 1

1 2 1 1 2 1

cos sin cos( )sin cos sin( )

cos cos( ) sin sin( )=

sin cos( ) cos sin( )

x y i

i

i

Ee

ee

δ

δ

δ

θ θ θ θθ θ θ θ

θ θ θ θ θ θθ θ θ θ θ θ

−− −

=−

− − −− + −

ur

After passing through the analyzer, the Jones vector is

1 2 1 1 2 1cos cos( ) sin sin( )0

i

x ye

Eδθ θ θ θ θ θ

−− − −

=ur

So the output light intensity distribution on screen is

2sin)(2sin2sincos

cos)sin(sin)cos(cos2)(sinsin)(coscos

)sin(sin)cos(cos

21212

2

121121

122

12

122

12

2

121121

δθθθθ

δθθθθθθθθθθθθ

θθθθθθ δ

−+=

−−−−+−=

−−−=∗= −

→

−

→i

yxyx eEEI

(1)

From formula (1) ,as 2 2πθ = (the direction of the polarizer is perpendicular to the analyze,) the light intensity will be

zero at the location of 1 0 θ = or2π

, the dark cross will be viewed observed(Fig.4 (a)), 2 0θ = (the direction of the

polarizer is parallel to the analyze,) the light intensity become strongest at the location of 1 0θ = or2π

, the bright cross

will be viewed(Fig.4 (b))[ 3].

(a) (b)

(a) the direction of the polarizer is perpendicular to the analyzer (P1⊥ P2) (b) the direction of the polarizer is parallel to the

analyzer (P1∥ P2) Fig. 4. Simulation light intensity distribution pattern of conoscopic light interference

2.2 Relationship between the rotation angle of the dark cross of the conoscopic interference pattern and the

linearly polarized light After derivation calculus to formula (1) , we can get ..

21 2 1 1 2 1

1

22 1

2sin [cos 2 sin(2 2 ) sin 2 cos(2 2 )]2

=2sin sin(2 4 )2

I δ θ θ θ θ θ θθ

δ θ θ

∂= − − −

∂

−

(2)

Let formula (2) equal to 0, we get

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2 12θ θ= (3)

we can reach the conclusion: if the polarizer rotation angle is θ 2,the angle of the cross shape will rotate θ 1=θ2/2 . It

is confirmed by Fig.4 , from Fig.4(a) to Fig.4(b) ,the direction of the polarizer P1 rotates π/2 , the dark cross rotates π/4.When the polarizer stand still, the linearly polarized light rotation angle is produced by optical activity substance, there is the same rule. We can take advantage of this rule to measure the rotatory angle : employ an area CCD camera to take a picture of the conoscopic light interference pattern before and after putting in the sample respectively. Comparing the two patterns and calculating the cross-shape rotate angle with the image processing software , the double angle is the rotatory angle.

3. CONOSCOPIC LIGHT INTERFERENCE PATTERN AND CALCULATION OF THE ROTATION ANGLE OF CROSS SHAPE

3.1 Experiment Setup

Figure 5 Schematic diagram of the experiment setup

Figure 6 : Experiment devices

Fig. 5 is the schematic diagram of the experiment setup. Fig. 6 are the experiment devices .Light source of the laser pass through polarizer, it becomes parallel linearly polarized light.. The optical activity substance rotates the linearly polarized light angle. When it passes the focusing lens, it becomes convergent beam, then oblique incidence into uniaxial crystal. After the analyzer it will form interference pattern on the ground glass, we use CCD camera to shoot the pattern, and use computer to deal with the image. 3.2 Conoscopic light interference pattern simulation

According to the formula(1) ，use the mathtools of MATLAB [10] to simulate the interference pattern with the corresponding of the light intensity to image gray level , depicture the pattern of conoscopic light interference. The parameters are as follow : wavelength 633nmλ = , the refractive index of BGO Crystal, 2.416en = , 2.363on = , it’s

thickness is 3mm , the distance between the crystal and the ground glass is 600mm , the maximum cone angle of incident light is arctan(1/ 60)α = ,the imaging range is [-10mm,+10mm], Figure 7 are the simulation patterns, we analysis them with image processing software to obtain the rotation angle of cross shape(bright or dark) in conoscopic interference pattern，make use of the expression 2 12θ θ= ， we can get the rotation angle of linearly polarized light .

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Figure 7 Simulation of conoscopic interference pattern with MATLAB ,

Calculating the angle of cross-shape pattern by image processing software , we obtain the angle of polarized plane rotation, The simulation data are list in Table 1 .

Table 1 Simulation data

the real rotation angle of the polarization light （°）

the pattern’s rotation angleθ 1（°）

Calculate the rotation angleθ 2

from the simulation pattern（°）

-90° -45.0006° -90.0011°

-70° -34.9995° -69.999°

-50° -24.9992° -49.9984°

-30° -15.0006° -30.0012°

-10° -5.0012° -10.0024°

0 -0.00075° -0.0015°

20° 10.00055° 20.0011°

40° 20.00065° 40.0013°

60° 29.99935° 59.9987°

80° 39.99915° 79.9983° 3.3 Conoscopic light interference patterns and calculating the rotation angle of cross shape In experiment ,we employ an area CCD camera to shoot the conoscopic light interference patterns as shown inFig.8(a),(b),corresponding to the direction of polarizer and analyzer is perpendicular or parallel with each other

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respectively . Employing the image processing algorithm to calculate the cross shape (the dark or bright cross) rotation angle from Fig.8 (a) to (b) , the angle is 44.6 o, the absolute error is 0.4o .

(a) (b)

(a) Image of cross shape as the direction of polarizer and analyzer is perpendicular with each other (p1⊥ P2)

(b) Image of cross shape as the direction of polarizer and analyzer is parallel with each other(p1∥ p2) Figure 8 Interference patterns of assembling polarization

The main measurement errors of this experiment arise from : a、 The light source intensity distribution is

nonuniform ；b 、There are noises in the pictures of the patterns to disturb image processing algorithm ; c、 The image of cone light interference is transformed into binary image,it will lose useful information .

4. CONCLUSION We propose a new method to measure the rotatory angle emerging as a linearly polarized light wave is passed through some optical activity substances by employing the conoscopic interference pattern. The principle of this method is explained, the rotation angle of the cross shape of the conoscopic interference pattern is one half of the rotatory angle. Using CCD camera to shoot the conoscopic interference pattern, calculate the angle of cross-shape pattern by image processing algorithm , we can obtain the angle of polarized plane rotation. We use this feature of conoscopic light interference to design experiment to measure the rotation angle. We believe that when the experiment device and the technique of image analysis is improved ,the accuracy will be improved . REFERENCES 1. LI Bin, TANG Zhe-fu .Research on Digital Technology for Polarmieter Measurement [J]. Control and Instrument in Chemical Industry, 2011,Vol.38(7):793-795

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