ジョーンズ計算法とニューラルネットワークを用いた偏光光ファイバ歪みセンサシステムに関する研究
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言語: English
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公開日: 2019-06-25
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キーワード (En):作成者: 趙, 耀
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https://doi.org/10.15118/00009901

Muroran Institute of Technology

Doctoral Dissertation

Study on polarization optical fiber
strain sensor system based on Jones
calculus and neural networks

Name: Yao Zhao
Supervisor: Assoc. Prof. Shinya Sato

Division of Engineering
IAbstract
Polarization sensors have attracted attention and been researched as they canbe constructed with a very simple experimental setup. They utilize the polarization
characteristic of light propagating in an optical fiber and determine the strain,stress, pressure, etc. from changes in the polarization state of light propagating
in a single-mode optical fiber when the sensor is affected by external factors.
However, in previous research, the use of either Hi-Bi fiber or polarization
maintaining fiber made the cost of the sensor system very high and operatedifficultly. In this work, we present a polarization sensor system that determines
strains by measuring changes in polarization state in a single-mode fiber which
are commercially available inexpensive fibers. This greatly reduces the cost of
the sensor system. In the experiment, we use the fiber wound in a coil shape asa sensor. When strains are applied, the sensor is rotated by the invar wire that is
attached to the sensor. Strains applied in the experiment can be determined fromthe measured polarization states when the sensor rotates.
In addition, we propose to calculate an arbitrary optical path using the Jones
matrices of two rotated wave plates. The calculated polarization states obtainedusing the proposed method are in good agreement with the measured values,
which demonstrated the feasibility of the proposed method. Therefore, when two
Jones matrices representing the optical path are obtained, the polarization stateof the transmitted light can be calculated from that of the reflected light on this
optical path and vice versa. Also, the strains applied in the experiment are in good
agreement with the calculated ones. These all verified the feasibility of the strainIIsensor system developed in this work. Therefore, when the object (building, slope,etc.) connected to the other end of the invar wire is deformed, the strain amount
can be obtained by this method. However, when a multipoint measurement isperformed by connecting sensors in series, the size of the experimental data
becomes enormous and it will become very difficult to use function fitting for dataprocessing. Taking that problem into account, we performed data processing
using a 3-layer feedforward neural network. The experimental data are used astraining data for learning, that is, we take the measured polarization state as the
input and the rotation angle of the sensor as the output for training the neuralnetwork. In addition, to test the trained neural network, we provided a newly
measured polarization state to the trained network. The output of the trained
network, namely, the rotation angle of the sensor, is almost the same as thatapplied in the experiment that generated the test data. These results demonstrate
the feasibility of both the sensor system and the data processing methods
proposed in the study.

III

Contents
1. Introduction .................................................................................................. 1
1.1 General Background ............................................................................... 2
1.2 Research Purpose ................................................................................... 3
1.3 Dissertation Outline ................................................................................ 5
2. Optical Fiber Sensor .................................................................................... 7
2.1 Fundamental of the Optical Fiber .......................................................... 8
2.2 Fundamental of Optical Fiber Sensors................................................ 10
2.3 Optical Fiber Sensing Method .............................................................. 122.3.1 Fiber Bragg Grating Sensor ........................................................... 122.3.2 Brillouin Optical Time Domain Reflectometry Sensor ................. 13
2.3.3 Polarization Optical Fiber Sensor .................................................. 14
3. Polarization Optical Fiber Sensor ............................................................. 16
3.1 Polarization ............................................................................................ 17
3.1.1 Types of Polarized light .................................................................. 17
3.1.2 Complex representation of electric field ....................................... 19
3.1.3 Stokes parameters .......................................................................... 20
3.2 Jones Calculus ...................................................................................... 21
3.3 Measurement Principle of the Sensor ................................................. 23
3.4 Measurement Principle and Problems in Multipoint Measurement .. 25
4. Neural Network ........................................................................................... 28
4.1 Outline of Neural Network .................................................................... 29
4.2 Components and Forms of Network.................................................... 30
4.2.1 Decisive 2-Value Model ................................................................... 31IV

4.2.2 Stochastic 2-Value Model ............................................................... 31
4.2.3 Decisive Analog Model ................................................................... 32
4.2.4 Architecture of Neural Network...................................................... 35
4.3 Backpropagation Method ..................................................................... 37
4.4 Design of Neural Network..................................................................... 42
5. Experiments and Results........................................................................... 46
5.1 Verification of twisting when rotating the sensor............................... 47
5.2 Experiment using Jones calculus and Results .................................. 51
5.2.1 Experiment system ......................................................................... 51
5.2.2 Experiment and results................................................................... 53
5.2.3 Changes in the polarization state due to rotation of the sensor. 61
5.3 Experiment using a Neural Network and Results ............................... 67
6. Conclusions ................................................................................................ 75
6. Conclusions ............................................................................................. 76
Acknowledgments.......................................................................................... 79
References ...................................................................................................... 80

V1. Introduction11.1 General Background
Natural disasters such as landslides and slope failures due to torrential rain,
typhoons, and earthquakes frequently occur in recent years [1-4]. Disruption oftransportation roads and collapse of houses due to such natural disasters has
become a social problem, and disaster countermeasures are strongly demanded[5-32]. Therefore, it is becoming increasingly important to monitor social
infrastructure such as roads, bridges, and tunnels in real time, to fully understandpre-disaster facts, to make correct decisions, and to take effective measures. Thisis expected to effectively reduce the damage caused by disasters. In addition,
disasters often occur in wilderness and mountainous areas where natural
conditions are severe.
Because of the vulnerability of these locations to lightning, rain, electromagnetic
interference, etc., traditional electrical sensors are often easily damaged and may
be difficult to monitor in real time. Since optical fiber sensors have features suchas light weight, small size, insulation, and electromagnetic induction resistance,
research for applying them to disaster prevention sensors is active. Since opticalfiber sensors have features such as light weight, small size, electrical insulation,and resistance to electromagnetic induction, research on sensors that make useof these features is actively conducted, and development of measurement
methods is in progress [31-51]. Among them, research for applying optical fiber
sensors to disaster prevention sensors is also becoming increasingly active. The
optical fiber sensor is not affected by electromagnetic interference because thesensor unit does not require a power supply. In addition, optical fiber sensors use
silicon dioxide, so they have high corrosion resistance. This is a very important2

aspect when setting up the sensor in the field especially where the environment
conditions are very harsh.
When an optical fiber sensor is used for measurement, the characteristics(phase, frequency, wavelength, intensity, polarization, etc.) of light propagating
through the optical fiber change when an external physical quantity is applied tothe sensor unit [52-53]. By measuring the change in the properties of the light, it
is possible to obtain an external physical quantity applied to the sensor. This isalso the measurement principle of the optical fiber sensor. There are various
types of optical fiber sensors depending on the measurement method. Amongthem, polarization sensors have attracted much attention and the developmentof the polarization sensors have been researched [54-61]. This is due to the factthat in addition to the properties of optical fiber sensors described above, thepolarization sensors can be constructed with a very simple experimental setup.
Therefore, there is a possibility that it can be applied to disaster prevention
sensing systems such as tunnel failure, earthquake prediction, and strain surveyof infrastructure.
1.2 Research Purpose
Optical fiber sensors have been widely used for sensing and monitoring invarious engineering fields because of their superiority to electrical sensors as
described previously. They are used to measure various physical quantities suchas strains, temperatures, pressures, current, vibration, displacement, and so on.
Among many types of optical fiber sensors, there is also an increasing number ofstudies on polarization optical fiber sensors that exploit the properties of the stateof polarization of light as a measurement method. This is because the polarization3

optical fiber sensors can be built with a simple structure compared with othersensing methods. When external influences are directly or indirectly applied tothe sensor, the polarization state of the light propagating in the fiber of the sensor
system changes. By measuring the change in the state of polarization, it ispossible to detect the external influences applied to the sensor. However, inprevious studies, polarization optical fiber sensor systems have used highbirefringence or polarization maintaining fibers for measurement [59]. The use of
these fibers not only makes the cost of the sensor system very expensive but
also makes it difficult to operate. These also limit to some extent the development
and spread of polarization optical fiber sensors. Therefore, we decided to develop
a small size, low-cost polarization fiber sensor system which can be used tomeasure strains. In the experiment, since a commercially available low-costsingle mode optical fiber is wound as a coil and used as a sensor, the cost is
greatly reduced compared to the conventional polarization optical fiber sensor.
In the experiment, we rotated the sensor by pulling an invar line connected tothe sensor. This is conducted to simulate the generation of strain with the
movement of the Invar line. That is, the strain is applied by pulling the invar line.
As the sensor rotates, changes in the state of polarization of light are measured
by the polarization analyzer. Since the measured polarization state corresponds
to the movement distance (strains) of the invar line, the distance of the invar line
can be determined from the measured polarization state. In this way, when thebuilding or the slope connected to the other end of the Invar line is deformed, thestrains of the measured object can be obtained by this measurement method.
Furthermore, the sensor system developed in this study is not affected by4external temperature changes and does not require temperature compensation.
In addition, the ultimate goal of this study is to connect multiple sensors inseries on a single fiber for multi-point measurement of strain. However, there is aproblem with multi-point measurement, which is the effect of the front-end sensor
on the rear-end sensor. Once the first sensor has rotated, the light entering thesecond sensor will change even if the second sensor does not rotate. Then, themeasurement result for the second sensor will have an error. To solve thisproblem, we propose to use the Jones matrices of two rotated wave plates torepresent an arbitrary optical path, so that we can calculate the polarization stateof the incident light entering the second sensor, even if the first sensor rotates. In
addition, in order to solve the problem of large data processing when themultipoint measurement is performed, we also propose the use of neuralnetworks for data processing. In this study, we use a three-layer feedforward
neural network for data processing.
1.3 Dissertation Outline
As described above, the introduction, i.e. Chapter 1, describes the generalbackground, research purpose, and dissertation outline.
Chapter 2 presents an overview of optical fibers and optical fiber sensors. The
concept, characteristics, and several sensing methods of optical fiber sensors aredescribed here.
Chapter 3 presents the outline of polarized light, Jones calculus, and themeasurement principle of polarization optical fiber sensors. In addition, we
describe the multipoint strain measurement principle based on the proposed
sensor system and the problems in multipoint measurement experiment.
5Chapter 4 describes the outline, components, and structure of neural networks.
In addition, the back propagation method, and the design of the neural networkin this study are also described in this chapter.
Chapter 5 presents the experiment and experimental results. The experiment
includes two parts. One is to verify the proposed method that an arbitrary optical
path can be computed using the Jones matrices of two rotated wave plates. The
other one is also the strain measurement experiment, however, this time we
performed the data processing using a 3-layer feed-forward neural network. The
experiment, data processing method, and the results are presented in this chapter,showing the feasibility of the proposed sensor system.
Finally, the conclusions drawn from this study are given in chapter 6.

6

2. Optical Fiber Sensor

72.1 Fundamental of the Optical Fiber
Optical fibers are flexible, thin thread-like transparent material made of quartz
glass or plastic, through which optical signals pass. Nowadays, optical fibers arebeing drawn to the home, making them indispensable for optical communications.
In addition, by taking advantage of the characteristics and structure of opticalfibers, they are also used for various applications such as sensors, fiber lasers,
high power light guides and the like [62-63]. The basic construction of an opticalfiber is a two-layer structure consisting of a core in the center and a cladding
material surrounding the core. The main component of the optical fiber corematerial is silicon dioxide, and a very small amount of dopants is used to increase
the refractive index of the core. In this way, the fiber core has a higher refractive
index than the cladding [64]. The general structure of an optical fiber is shown in
Fig 2.1.1.

Core with higher
refrective index

Cladding with lower
refrective index

Fig.2.1.1. Structure of an optical fiber.
As previously described, the fiber core is doped with impurities in order toincrease the refractive index of the core. This is done to take advantage of thetotal reflection of light in order to confine the optical signal inside the core of the8fiber for long distance transmission. As long as we adjust the angle at which lightis incident into the core of the fiber, we can ensure that light is totally reflected at
the boundary of the fiber core and the cladding and is transmitted through thefiber. Total internal reflection is also a necessary condition to transmit opticalsignals using an optical fiber. Only then can the optical signal be completely
confined within the core of the optical fiber for propagation, thereby avoiding
unnecessary signal losses. It is necessary to ensure that optical signals
propagate within the fiber core not only in the field of communications but also in
the field of fiber sensing. Typically, the core has a refractive index of 1.47 and the
cladding has a refractive index of 1.46. The propagation of the optical signal in
the core of the fiber using total internal reflection is shown in Fig.2.1.2 [65].

Cladding
Core

Cladding

Fig. 2.1.2. Propagation of light in the fiber core.
Optical fibers are roughly classified into multimode optical fibers (MMF) andsingle-mode optical fibers (SMF) depending on the propagation mode. Multimodeoptical fiber is one designed to carry from hundreds to thousands of light rays or
propagation modes, each at a slightly different reflection angle within the optical9

core. Multimode optical fibers generally have a wider core diameter, typically 50100 micrometers. And multimode optical fibers are mostly used forcommunication over relatively short distances because of the large inter-mode
dispersion of multimode fiber, resulting in narrow fiber bandwidth. On the other
hand, single mode optical fiber has only one propagation mode, and it has nointer-mode dispersion, so the bandwidth is very wide. Therefore, it is best to use
single-mode fiber with very wide bandwidth for high speed and long distancetransmission systems. In this study, single-mode fibers are used to implement theexperiments.
2.2 Fundamental of Optical Fiber Sensors
In recent years, with the development of optical fiber communication
technology, laser light source technology, and optical fiber sensing technology,
optical fiber sensors have been vigorously researched and developed in the past
decades. One important reason is that the fibers that make up the fiber optic
sensor have many unique features and advantages, which are described asfollows [66]:(a) thin diameter(b) lightweight(c) flexibility(d) non-inductive(e) corrosion resistance(f) explosion proof
These features are very useful for installation and construction even in a narrow
space that is not easily accessible, and in transportation equipment where
10exercise performance and transport efficiency are important. Although traditional
electrical sensors are widely used in many fields, the measurement unit always
requires a power supply which is susceptible to lightning and surge current. Thisis a serious problem that is urgent to be solved. In addition, if there are electronicand electrical devices nearby, sensors and measures that do not receive
electromagnetic induction are required. The advantages of the aforementioned
optical fibers have enabled fiber optic sensors to successfully solve the problems
faced by these electrical sensors.
Nowadays, optical fiber sensors are playing an increasingly important role inmodern life and industry and are currently being studied to be more sensitive,
more precise and more adaptable. They have been widely used for measuringphysical, biological, and chemical parameters in various fields. An optical fiber
sensor measures external changes by detecting changes in light properties suchas intensity, phase, frequency, and polarization. The optical fiber sensor basicallyconsists of a light source, an optical fiber as a transmission path, measurement
equipment and an optical receiver. However, from the specific role of the fiber inthe sensing system, the sensor is generally divided into two types [67]. That is tosay, in the fiber sensor system, the fiber is used only as a transmission path, orthe fiber itself is used as a sensor. The first is that there are sensor components
in the sensing system, and the fiber is only used as a transmission path. The
second is that the fiber itself is used as a sensor, and the external action is applied
directly to the fiber. In this research, we conduct experiments with the optical fiber
itself as a sensor. This kind of optical fiber sensor has very unique features andcharacteristics, which include: small size, lightweight, flexibility, immune to11electromagnetic induction, corrosion resistance, and distribution and quasidistribution measurement [31-32]. These characteristics and features allow theuse of fiber optic sensors in places where traditional electrical sensors cannot
apply, such as the places where there is electromagnetic interference or where
thunderstorms occur frequently. In this study, the state of polarization of lightpropagating in optical fibers will change when external changes are applied tothe sensor, therefore, we can measure the changes in the state of polarization oflight to determine the external changes that are applied to the sensor.
2.3 Optical Fiber Sensing Method
There are various methods of fiber sensing, and they all have their own
characteristics. When we want to choose a sensing method, we need to have agood understanding of how this method works. On the basis of understanding theprinciple and selecting for your own measurement target, satisfactorymeasurement results will be obtained. Here, we mainly introduce fiber sensing
methods based on three measurement principles. The first is the fiber Bragg
grating (FBG) detection method that reflects a specific wavelength, the second is
the Brillouin Optical Time Domain Reflectometry (BOTDR) detection method
using backscattered light, and the third is the polarization detection method usedin this study.
2.3.1 Fiber Bragg Grating Sensor
FBG is made by irradiating a doped fiber with a strong laser to change therefractive index of the fiber core. The refractive index of the fiber will change
correspondingly with the spatial distribution of the luminous intensity. The fibercore after strong laser irradiation has a periodic change in refractive index and
12becomes a fiber grating. An FBG reflects only a specific wavelength called the
Bragg wavelength λ𝐵 and transmits the other wavelengths when lthe ight isincident to the FBG [34]. The Bragg wavelength λ𝐵 can be expressed by Eq.
(2.3.1).
𝜆𝐵 = 2𝑛𝛬(2.3.1)In the equation, 𝑛 is the effective refractive index and Λ is the grating period.
When strain or temperature is applied to the FBG, both n and λ change and the
Bragg wavelength shifts. Since the shift of the Bragg wavelength has a linear
relationship with the change in temperature and strain, the temperature and strain
applied to the FBG can be detected by observing the shift of the Bragg
wavelength.
As described above, an FBG sensor is not only sensitive to changes intemperature but also sensitive to temperature changes. It is therefore often used
as a strain sensor and temperature sensor to measure strain and temperaturechanges. This also means that temperature compensation must be performed
when measuring strain using FBG sensors.
2.3.2 Brillouin Optical Time Domain Reflectometry Sensor
When an optical pulse is incident on an optical fiber, it generates various
scattered light. The scattered light that returns to the light incident end is calledbackscattered light. There are various types and characteristics of scattered light,
one of which is called Brillouin scattered light depending on strain andtemperature changes. Brillouin scattered light has a feature that the frequency
distribution of Brillouin scattered light generated in that part changes due to strain
or temperature change applied to the optical fiber. That is, when the strain or13

temperature change is applied to the optical fiber, the frequency of the Brillouinscattered light will shift. Since the Brillouin frequency shift amount changes
linearly with respect to strain and temperature, measuring this shift amountenables detection of strain or temperature change applied to the optical fiber [68].
In this method, since the optical fiber itself is a sensor, linear and planarmeasurement and monitoring that cannot be realized by conventional
measurement have become possible. Another advantage of the BOTDR
measurement system is that it can measure continuous distribution for several
tens of kilometers from one end of a single optical fiber [69]. Furthermore, if thesensor optical fiber is installed in a loop, measurement can be continued bymeasuring from both ends even if there is a break at one place.
However, since the Brillouin scattered light signal returning to the incident end
is low, it is necessary to perform averaging processing. Therefore, it takes tensof minutes for one measurement. This also means that real-time dynamic
measurements are not possible with BOTDR sensor systems.
2.3.3 Polarization Optical Fiber Sensor
The polarization optical fiber sensor is a sensor that detects the amount ofstrains by measuring the changes in the polarization states when strains areapplied to the sensor. In principle, it is possible to connect multiple sensors on
one fiber, so multipoint measurement can be implemented with one opticalmeasurement instrument. Slope installation is simpler than conventional types
because polarization is less susceptible to loss of fiber connections andconnectors can be used for installation and extension. Therefore, it is expected
to reduce installation cost. In this study, the fiber used as a sensor does not14require any special treatment, and the experiment can be completed using
commercially available single-mode fiber. Compared to studies using birefringent
fibers or highly polarization maintaining fibers, we can achieve lower prices. The
detailed description of polarization optical fiber sensors will be presented in
Chapter 3.

15

3. Polarization OpticalFiber Sensor

163.1 Polarization
Light is an electromagnetic wave whose vibration direction is perpendicular tothe propagation direction. Light in which an electric field and a magnetic field
vibrate only in a regular direction in this plane is called polarization. Random lightwith no regularity in the polarization of light waves is called natural light [70]. The
polarized light can change shape freely without losing energy or increasing
entropy. Also, since polarized light is the simplest light, it can be handled
physically briefly, and mathematical handling for experimental prediction can beeasily done [71-72]. From such a perspective, polarized light is easier to handle
than natural light. Cross-sectional views are often used to represent polarized
light. The sectional view is the trajectory of the tip of the electric field vector when
the observer stands on the path of the light ray and looks toward the light sourceat the origin of the coordinate.
3.1.1 Types of Polarized light
When considering the electric field of plane polarized light traveling in the Z
direction, let each component 𝐸𝑥 and 𝐸𝑦 be expressed as:

E x  Ax cos(t  kz   x ) E y  Ay cos(t  kz   y ) 

(3.1.1)The polarization state is determined by the amplitude ratio Ax ⁄Ay and thephase difference 𝛿 = 𝛿𝑦 − 𝛿𝑥 [73].
(a) Elliptically polarized light
If 𝐴𝑥 ≠ 𝐴𝑦 and it is an arbitrary phase difference δ which is shown in Eq.
(3.1.1), the locus of the tip of the electric field vector is

17

2

E E
 Ex   E y   2 x y cos   sin 2   
Ax Ay
 Ax   Ay 
2

(3.1.2)and it will draw an ellipse. This polarized light is called elliptically polarized light,
and the cross-sectional view is shown in Fig. 3.1.1 (a). The inclination angle α
between the major axis of the ellipse and the x-axis is shown as

tan 2  2

Ax Ay
Ax  Ay
2

2

cos 

(3.1.3)

The rotation direction of the electric field is counterclockwise when 0 <δ <π,
and clockwise when −π<δ <0. Common polarized light is elliptically polarized lightand it becomes circularly polarized light and linear polarized light when special
conditions are met.
(b) Circularly polarized light
If 𝐴𝑥 = 𝐴𝑦 and cos 𝛿 = 0 in Eq. (3.1.1), the polarized light is called circularpolarized light. The cross-sectional view is shown in Fig. 3.1.1 (b). The
determination of the direction of rotation is the same as in the case of ellipticallypolarized light(c) Linearly polarized light
If 𝐴𝑥 = 𝐴𝑦 and cos 𝛿 = 1 in Eq. (3.1.1), the polarized light is called linearlypolarized light. The cross-sectional view is shown in Fig. 3.1.1 (c). The inclination(orientation angle ) from the x-axis of the oscillating plane of the electric field isshown as

 Ey 
E x

  tan 1 (3.1.4)

18

y

y

E

y

E

α

ϕ
x

x

(a) Ellipticallypolarized light(b) Circularly
polarized lightx(c) Linearlypolarized light

Fig. 3.1.1. Types of Polarized light
3.1.2 Complex representation of the electric field
Since the equation of the electric field can be expressed in the complex form,
Eq. (3.1.1) can be rewritten to the complex form as [73]Ex  Axei (t  kz )E y  Ay ei (t  kz  ) 

(3.1.5)

Shifting the phase reference − 𝛿 ⁄2






i (t  kz   ) 
2 E y  Ay e

E x  Ax ei (t  kz  )2(3.1.6)Since the polarization state is determined by the ratio of the amplitude and thephase difference, spatial and temporal changes can be ignored. Eq. (3.1.6) canbe represented as
E x  Ax e 2 iE y  Ay e 2 i(3.1.7)19

3.1.3 Stokes parameters
In the previous section, the polarization state is expressed using the amplitude
and phase of light, but it is not easy to measure the amplitude and phase of lightwhen measuring the polarization state. When actually making measurements,
use Stokes parameters as light intensity. The Stokes parameters consist of fourterms S0 , 𝑆1, S2 and S3 . From S0 , the intensity of the entire light, the intensityof the horizontal linear polarization component, the intensity of the + 45 ° linear
polarization component, and the intensity of the right circular polarization
component are respectively shown. If temporal changes of the polarization state
can be ignored and the complex conjugate is * [73-74], the Stokes parameter canbe defined as

S 0  E x E x*  E y E *y  Ax2  Ay2
S1  E x E x*  E y E *y  Ax2  Ay2
S 2  E x E *y  E y E x*  2 Ax A y cos 
S 3  E x E *y  E y E x*  2 Ax A y sin 









(3.1.8)Completely polarized light is shown as(3.1.9)S 02  S12  S 22  S32

In many cases, standardized Stokes parameters are used in which eachpolarization component of 𝑆1 to S3 is standardized by the overall intensity S0 .
The phase difference can be obtained by Eq. (3.1.10).
S𝛿 = tan−1 ( 3)(3.1.10)S2The Stokes parameters and the electric field representations are interrelated
and can be converted to each other by Eq. (3.1.8) and Eq. (3.1.10). In thisexperiment, the polarization state is handled by converting the Stokes

20

parameters obtained by the polarization analyzer into the electric field display.
3.2 Jones Calculus
As described above, polarization represents a state in which the electrical field
vibration of a plane of light is spatially biased [73-74]. When the electrical field isrepresented by 𝐄, then the arbitrary polarization can be represented by a vector
sum of two fundamental polarized lights 𝐸𝑥 and 𝐸𝑦 The parameters fordetermining the polarization state is the phase difference 𝛿 of 𝐸𝑥 and 𝐸𝑦 , andthe ratio of the amplitudes of 𝐸𝑥 and 𝐸𝑦 .
There are two methods for calculating the polarization of an optical path, theJones matrix calculus, and the Mueller matrix calculus [75-76]. Since completely
polarized light is used in the experiment, for each optical element we decided touse the Jones matrix as it is easier to calculate than the Mueller matrix. The Jones
matrix calculus is a method of representing polarized light by a matrix. It is
composed of a 2⨯1 Jones vector which represents an electrical field component,
and a 2⨯2 Jones matrix representing the polarization component. Using complex
notation, the 2⨯1 Jones vector representing the electric field is expressed as:  x   0 x exp{i ( wt  kz   x )}
 0 x exp(i x ) Ε 
 exp[i( wt  kz )] 
 (3.2.1)
 y  0 y exp{i ( wt  kz   y )}
0 y exp(i y ) For actual measurements, only the phase difference is taken into account, so
Eq. (3.2.1) can be rewritten using a phase difference while omitting exp[i(wt  kz)]as shown in Eq. (3.2.2) [32].
0 x 0 x exp(i x )  

Ε



0 y exp(i y )  0 y exp(i ) (3.2.2)   y x(3.2.3)21

where E0𝑥 and E0𝑦 are amplitudes, 𝛿𝑥 and 𝛿𝑦 are phases, i=√−1, and 𝛿 isthe phase difference. A polarization element is represented by a 2⨯2 Jones
matrix 𝐉, which is shown as:jJ   11
 j21

j12 
j22 

(3.2.4)

When the optical element receives an incident light represented by the Jones
vector 𝐄𝒊𝒏 , it changes the polarization state of the incident light to emit a newpolarization state vector 𝐄𝒐𝒖𝒕 . The polarization conversion process by the opticalelement is expressed by [32]:𝐄𝑜𝑢𝑡 = 𝐉 · 𝐄𝑖𝑛(3.2.5)In addition, when there are M optical elements, we can use Eq. (3.2.6)
expressed below to calculate the transmitted light through multiplying the Jones
matrices representing the optical elements in order from right to left [32].

𝐄𝑜𝑢𝑡 = 𝐉𝑀−2 · 𝐉𝑀−1 · 𝐉2 · 𝐉1 · 𝐄𝑖𝑛(3.2.6)

However, in practical applications, we can only use this method after obtaining
the Jones matrices of all optical elements in the optical path. It is often very
difficult to obtain the Jones matrices of all optical elements in the optical path.
Therefore, we propose to use the Jones matrices of two rotated wave plates tocalculate an arbitrary optical path. Thus, no matter how many optical components
are included in the optical path, we can determine the polarization state of thetransmitted light from the polarization state of the incident light of the optical path
as long as we can find the Jones matrix of the two rotated wave plates
representing the optical path. This calculation can be expressed by Eq. (3.2.7).

22

In addition, the Jones matrix of an arbitrarily rotated wave plate is expressed by
Eq. (3.2.8) [77], which is a regular matrix.

𝐄𝑜𝑢𝑡 = 𝐉2 · 𝐉𝟏 · 𝐄𝑖𝑛cos 2  i sin 2 cos 2
J 2 , J1   i sin  cos 2
2(3.2.7)
(3.2.8)
cos  i sin cos 2 22
i sin cos 2
2where ∅ represents the phase difference of the arbitrarily rotated wave plate,and 𝜃 represents the orientation angle.
3.3 Measurement Principle of the Sensor
As shown in Fig. 3.3.1, the sensor that is used in the experiment is a singlemode 8.3/125 fiber that is wound in a circle shape [31]. When the fiber is wound
into a circle, the core and cladding of the fiber are usually deformed, thereby
changing the fast axis refractive index and the slow axis refractive index.
Birefringence in the fiber is caused by changes in the fast and slow axis refractive
indices. The influence of this birefringence can be increased depending on thesize of the winding radius to be integrated from end to end of the wound portionof the optical fiber. By adjusting the number of turns and the winding radius of the
optical fiber, it is possible to have the same performance as a wave plate. In other,
the circularly wound fiber can be used as a wave plate. The phase difference,
which is expressed as 𝛿, of the wound part can be calculated by [31]:
=

m

2

(3.3.1)m

 R
2 ar 2 N

(3.3.2)where N is the number of turns of the fiber; r is the cross-sectional radius of the23

fiber; R is the bending radius of the fiber; λ is the wavelength of the laser light;
and a is calculated to be 0.13 [78]. In the experiment, N is set to 1, R to 0.0215
m, r to 6.25⨯10-5 m, and λ to 1.55⨯10-6 m. Then we substituted these values into
Eq. (3.3.1) and (3.3.2) and calculated the phase difference of the sensor to be
34.43°. The orientation angle of the sensor was the actual rotation angle of thesensor applied in the experiment.

R

r

(a)

A
B
C
D
Invar wire(b)
Fig. 3.3.1. Schematic diagram of the sensor.
Figure 3.3.1(b) shows the actual photo of the sensor part, including a top view
and a left view. The base (A) with a sector part (B) is rotatable, and the radius of24

the sector part is donated as 𝑅 ′ . And, we fixed the cylinder (C) with a radius of
0.02m to the base using screws. Then, the optical fiber wound around thiscylinder was used as a sensor and is fixed in the fixing slots (D) [32]. In theexperiment, the Invar wire was attached to the sector part. When we pull the Invar
wire to rotate the base, the sensor will also rotate with it. Then, we can use therotation angle of the sensor to calculate the movement distance of the Invar wire,and further determine the strains applied in the experiment. Since the rotation
angle of the sensor is obtained from the measured polarization state in theexperiment, determining the strains applied through measuring the polarization
state is possible. Here, the strain, which is denoted as ε, is defined as thepercentage ratio between the movement distance ΔL of the invar wire and thelength L (1 m) of the invar wire, which are expressed as [32]L   R '




180

,

(3.3.3)L,
L(3.3.4)

where R ' is 0.065 m and 𝜌 (∘) is the rotation angle of the sensor.
3.4 Measurement Principle and Problems in Multipoint Measurement
In practical applications, there are many occasions where multipoint monitoringand measurement of strains is required. Based on the strain measurement
system proposed in this study, multipoint measurement of strains can be
performed using the time delay of reflected pulses from sensors when they areconnected in series. However, in multi-point measurement, there is a problem
that the front-end sensor affects the rear-end sensor. However, in multipoint
measurement of strains, the problem arises that the front-end sensor affects the25rear-end sensor. Here, we take the two sensors connected in series as an
example to illustrate the problems encountered in multipoint measurement. Whenthe first sensor rotates, the polarization of the transmitted light from it changes,
which directly causes the polarization of the incident light entering the second
sensor to change. This also causes the polarization state of the reflected lightfrom the second sensor measured in the experiment to change, even if thesecond sensor does not rotate. In addition, in the multipoint strain measurement
experiment, once all the sensors are connected, we cannot separately measure
the transmitted light from each sensor or the incident light entering each sensor.
The polarization states of the reflected light from each sensor are the only
parameters we can measure. If we can calculate the polarization state of theincident light incident entering the second sensor, and based on the measured
polarization state of the reflected light of the second sensor, we can correctly
obtain the rotation angle of the second sensor. That is, if we can correctly
calculate the polarization state of the incident light entering each sensor in themultipoint experimental system, multipoint strain measurement can be
implemented. However, in the present experiment, the polarization state of theincident light of each sensor could not be correctly calculated by the original
calculation method. Therefore, as mentioned earlier, we propose to use the Jones
matrices of two rotated wave plates to represent an arbitrary light path. Then,regardless of whether the first sensor is rotated, we can correctly determine thepolarization state of the incident light entering the second sensor as long as wefind the Jones matrices representing the optical path. The calculated value of thepolarization state of the incident light entering the second sensor is then used26

instead of its actual value. In this way, we believe that the problem of the influenceof the first sensor on the sensor sensor in multipoint strain measurement can besolved. This method is also applicable when there are three or more sensors. As
can be seen from the foregoing, when the sensor is rotated, we measure thepolarization state of the reflected light from the sensor, represented by three
Stokes parameters. That is, one rotation angle will correspond to three Stokes
parameters. In the multipoint strain measurement experiment, two or moresensors will be connected, and the experimental data will become very large. It
will take time to calculate the rotation angle of the sensors from the polarization
states of each sensor because the polarization state of the reflected light of eachsensor cannot be calculated at the same time. Moreover, an error has alreadyoccurred in the calculation of the polarization state, and it is possible to generatea larger error when using the calculated values of the polarization state to obtain
the rotation angle of the sensor. Therefore, we use neural networks for dataprocessing in our research because neural networks have the ability to handle
multiple inputs and multiple output problems. As long as we design the neuralnetwork we need, we can get the rotation angle of each sensor at the same time.
This not only saves a lot of time but also improves the accuracy of dataprocessing. The detailed description of the neural network will be introduced in
Chapter 4.

27

4. Neural Network28

4.1 Outline of Neural Network
A neural network is a mathematical model established by abstracting thehuman brain neuron network from the perspective of information processing. In
engineering applications, it is commonly referred to as an artificial neural network.
Neural networks can effectively approach a wide range of applications. In
particular, it exhibits excellent capabilities in the problems of pattern mapping,pattern perfection, pattern classification and the like. The first research on neuralnetworks information processing is the model of neurons proposed by McCulloch
and Pitts in 1943 [79]. They described the concept of neurons in their research,which are the units that exist in the network, mainly accepting input, processing
input, and producing output. A neural network is a network constructed bymutually connecting artificial elements simulating nervous system cells (neurons)of a living body, and has three major features [80-82].
The first characteristic is "nonlinearity". It is well known that neural networks
have powerful approximation capabilities and can achieve high precision
approximation for any complex function. The reason for this ability is the nonlinear
nature of the network. With the linear activation function, the neural network
simply combines the input and outputs it again, so the approximation of complex
functions cannot be achieved. With nonlinear activation functions, neuralnetworks can approach complex functions at will.
The second characteristic is "learning ability", which is also said to be thegreatest attraction of neural networks. Learning ability is the ability toautomatically form the necessary functions based on the examples presented.
This ability is suitable for problems that cannot be solved by conventional29

methods. In a system with learning ability, it is not necessary to design themechanism artificially.
The third feature is "parallelism". A neural network is one of the realizations ofmassively parallel computers, and a theoretical system for systematicallygrasping parallel operation has been established.
4.2 Components and Forms of Network
A neural network is a network that simulates the human brain neurons in orderto realize artificial intelligence-like machine learning technology. It is a model
established from the abstraction of specific functions of the human brain neuron
network from the perspective of information processing. A neuron model is amodel that contains inputs, outputs, and computational functions, much simpler
than the actual neurons of a human brain. In a neural network, a neuron is thebasic unit of computation. It accepts external inputs or inputs from other neurons
and produces outputs after computation. There are several different models ofneurons depending on the method of computation, and we need to choose theappropriate model based on the characteristics of our experimental data. When
deciding the model of a neuron, it is generally considered from two aspects [80].
The first one is the operation of the element is " decisive " or "probabilistic". The
other is whether the output is "2 -value" or "analog". Currently, neural network
engineering models that most used in neural networks include decisive 2-value
models, stochastic 2-value models, and decisive analog models. In our
experiments, the polarization state is an analog value that is critical to the rotation
angle. Therefore, this study uses a "Decisive analog model". The neuron modeland the activation function used are described as follows.

30

4.2.1 Decisive 2-Value Model
The decisive 2-value model is modeled by a decisive element with a 2-value
output of {0, 1} corresponding to whether the neuron is excited or not [80-82].
We assume the signal input to the model to be {x1, x2 … xN}. By multiplying
these inputs by weights {w1, w2 … wN} and adding them together, an amount s
corresponding to the intracellular potential is obtained. Here, the weights
correspond to the synaptic connection weights. If s exceeds the threshold θ, theoutput y will be 1, otherwise, the output y will be 0. The calculation is shown in
Eq. (4.2.1) and Eq. (4.2.2). And, the decisive 2-value model is shown in Fig. 4.2.1.
s = ∑𝑁𝑛=1 𝑤𝑛 𝑥𝑛(4.2.1)1y={
0(4.2.2)

𝑠≥𝜃
𝑠≤𝜃

input

y

s

f

output
y

1
0 θ
+

s
+ +

Fig. 4.2.1 Decisive 2-value model.
4.2.2 Stochastic 2-Value Model
A stochastic 2-Value Model is a model to decide stochastically whether to set this
output valued to either {0, 1}, which is shown in Fig. 4.2.2. That means even if the
input is the same, the output may take different values [80-82].
31

inputpsf+

output
y

1
0.5

0 θ

s

+ +

Fig. 4.2.2. Stochastic 2-value model.
The intracellular potential s is calculated from the input in the same way as thedecisive model. Then, calculate the sigmoid function value p for s, and set theoutput y to 1 with the probability of p, and 0 with the probability of 1-p. The
calculation is expressed by Eq. (4.2.3) to Eq. (4.2.6) [80-82].
s = ∑𝑁𝑛=1 𝑤𝑛 𝑥𝑛(4.2.3)p = sigmoid(s − θ)

(4.2.4)P(𝑦 = 1) = p(4.2.5)P(𝑦 = 0) = 1 − p(4.2.6)4.2.3 Decisive Analog Model
A decisive analog model is a model that determines the output y as an analog
value in the range of 0 ≤ y ≤ 1 and definitely determines this output from the
input. The intracellular potential s is obtained by the same procedure as the 2value mode described above. However, different from the decisive 2-value model,
the continuous output will be obtained by using a function called a sigmoid

32

function when calculating y from s. The sigmoid function is shown in Fig. 4.2.3,
and its equation is expressed as Eq. (4.2.7) [80-82].

Fig. 4.2.3. Sigmoid function.
1

sigmoid(z) = 1+𝑒 −𝛼𝑧(4.2.7)

It is an important function representing input/output characteristics of neurons.
In Eq. (4.2.7), α is a coefficient which is called gain. The output of the Sigmoid
function is a continuous value, and the Sigmoid function is a monotonic function
for the input, usually used for backpropagation method. The sigmoid function andits derivative are also continuous. Also, as shown in Fig. 4.2.3, the sigmoid
function approaches the step function if the gain of the sigmoid function isincreased. Conversely, when the gain is larger than 0 and sufficiently smaller than
1, the sigmoid function becomes a linear function. This requires us to set the initialvalue of the gain, neither too large nor too small, but to choose a suitable initialvalue.
As shown in Fig. 4.2.3, the function rises with z=0 as a boundary, and when
the threshold is θ, the function can be shifted to the right by θ, and the function

33

can be rewritten as(4.2.8)y = sigmoid(s − θ)The decisive analog model is shown in Fig. 4.2.4, and the output of a decisive
analog model can be calculated using Eq. (4.2.9) and Eq. (4.2.10).
s = ∑𝑁𝑛=1 𝑤𝑛 𝑥𝑛(4.2.9)y = sigmoid(s − θ)

(4.2.10)

input
y

s

f
+

output
y

1

0 θ

s

+ +

Fig. 4.2.4. Decisive analog model.
As can be seen from Fig. 4.2.3, the decisive analog model has two kinds ofparameters, the weights and the threshold. During the learning process, we
modify the weights and thresholds to make the output of the network to approach
our expected target output. Therefore, the fewer the number of parameters, theeasier it is to modify. In order to unify the parameters for processing, we usually
make some changes to the neurons as shown in Fig.4.2.5. It adds an input x0 ,which takes a value of 1 to the input. Let the value of weight w0 be -θ for this
input. This means that the threshold θ can be treated as one of the weights. The
34

intracellular potential s and the output 𝑦 of the neuron can be expressed by thefollowing equation [83].

s = ∑𝑁𝑛=0 𝑤𝑛 𝑥𝑛(4.2.11)𝑦 = 𝑓(𝑠)(4.2.12)

input
Activation
function

s

f

y
1

output
y

0 θ

+

s

+ +

Fig. 4.2.5. Treat threshold as one of weights.
4.2.4 Architecture of Neural Network
As mentioned earlier, neural networks are constructed by interconnecting
neurons that simulate human brain neurons. In practical applications, neuralnetworks are generally divided into two categories. One is a feedforward neuralnetwork and the other is called a feedbackward neural network [81-83].
(1) Feedforward neural networks
The feedforward neural network consists of a series of layers, including an
input layer, several hidden layers, and an output layer. The first layer, called the
input layer, receives signals from the outside as input to the network. A simple
schematic of a feedforward neural network is shown in Fig. 4.2.6.

35

inputoutputinput

output
・
・
・

・
・
・

・
・
・

・
・
・

input

outputInput layer

hiddend layer

Output layer

Fig. 4.2.6. An example of a feedforward neural network.
Each of the subsequent layers is connected to the previous layer, and the lastlayer produces the final output of the network. Inside the network, the signals
propagate in the order of the input layer, the hidden layer, and the output layer,
and are processed by the neurons at each layer, and the final operation result is
outputted by the output layer. There is no signal feedback inside a feedforward
neural network.
(2) Feedbackward neural networks
Unlike a feedforward neural network, within a feedback neural network, thesignal flow is not unidirectional, but bidirectional. When determining the neurons
on the input side, the values of the neurons on the output side must be consideredat the same time. At the same time, the value of the neurons on the output side
also depends on the value of the neurons on the input side. Figure 4.2.7 shows
an example of a feedbackward neural network. We find that it is possible toexchange signals between arbitrary neurons. This makes the feedback neuralnetwork look very complicated.

36

output

input

input

Fig. 4.2.7. An example of a recurrent neural network.
Since the signal is fed back from the output side to the neurons on the input
side, the feedbackward type neural network has complex nonlinear dynamics andexhibits various time characteristics, which can be used to solve dynamic
information processing [83].
4.3 Backpropagation Method
The error back propagation method is one of the learning methods offeedforward neural networks, and has been widely studied and actively used inneural network algorithms. The error back propagation method determines theparameters of the network based on the gradient method [82-85]. The error
backpropagation network has three or more layers, but basically, it is usually
three layers.

37

・
・
・

input layer

・
・
・

・
・
・

hidden layers

・・・

output layer

Fig. 4.3.1. Feedforward neural for error backpropagation method.
As shown in Fig. 4.3.1, there are known inputs ( 𝑥1 , 𝑥2 , … 𝑥𝑁 ) andcorresponding target outputs (𝑡1 , 𝑡2 , … 𝑡𝑁 ). In the process of learning with neuralnetworks, the neural network will give outputs (𝑦1 , 𝑦2 , … 𝑦𝑁 ). The purpose of ourstudy is to make the error between the outputs ( 𝑦1 , 𝑦2 , … 𝑦𝑁 ) of the neuralnetwork and the target outputs (𝑡1 , 𝑡2 , … 𝑡𝑁 ) infinitely close to 0. We use the error
evaluation function 𝐸 to calculate the error between the outputs (𝑦1 , 𝑦2 , … 𝑦𝑁 )and the target outputs (𝑡1 , 𝑡2 , … 𝑡𝑁 ), which is given as [83]2𝐸 = ∑𝑁𝑛=1|𝑦𝑛 − 𝑡𝑛 |(4.3.1)In this study, the learning method we used was the backpropagation method.
Change the relationship between inputs and outputs by changing the internalparameters (weights and thresholds) of the network.
Here, we take the feedforward neural network shown in Fig. 4.3.1 as an
example. The neurons that make up the neural network are the neuron models
we described in Section 4.2.3. In this case, the threshold is regarded as one of
the weights, and only the weight of the neurons needs to be modified during the38

learning process.
In this study, in order to process the experimental data, we used a three-layer
feedforward neural network. Therefore, we are here to explain the error backpropagation method based on the feedforward neural network. The measured
values of the three parameters of the Stokes representing the polarization state
are used as inputs, and the angle of rotation of the sensor is used as an output
to the network. When designing neural networks, we must consider our actual
data types and structures. For our experimental data, what we need is a threeinput-one-output neural network, as shown in Fig.4.3.2. Thus, we give theintroduction of the error backpropagation method using the neural network shownin Fig. 4.3.2.

=1=1

Fig.4.3.2. Forward mode of error backpropagation method.
In the learning process of the error back propagation method, it is divided into

39

two steps, one is the forward mode and the other is the backward mode. The stepfor calculating the actual output of the neural network in the order of the input
layer to the output layer is called the forward mode, as shown in Fig.4.3.2. Here,the 𝑥1 , 𝑥2 and 𝑥3 are inputs to the neural network, and the 𝑦1 is the output of
the neural network. The 𝑤 represents the weight, and the threshold of eachneuron is treated as a weight. So we add a neuron with an output of 1 and athreshold of– 𝜃 in each layers.

Before learning begins, we need to give each weight an initial value. The input
and target output are then provided to the neural network and learning begins.
Learning is actually to determine a set of weights that correctly represent therelationship between input and target output. When the learning is complete, theset of weights will be determined. The actual output of the neural network can becalculated according to the equations as follows:
𝑆11 = 𝑤11−1 𝑥1 + 𝑤21−1 𝑥2 + 𝑤31−1 𝑥3 + 𝑤01−1 𝑥0(4.3.2)(4.3.3)𝑦𝑠11 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑠11 )𝑦1 = 𝑤11−2 𝑦𝑠11 + 𝑤21−2 𝑦𝑠21 + 𝑤31−2 𝑦𝑠31 + 𝑤01−2 𝑠0(4.3.4)𝑦𝑠11 , 𝑦𝑠21 and 𝑦𝑠31 are the output of neurons in the hidden layer and are also
the input to the output layer. In a neural network, the output of each layer is the
input to the next layer connected to it. Therefore, we can determine the output of
neurons of each layer in the order from the input layer to the output layer.
The next step is the backward mode that corrects the weights of each layerfrom the output layer to the input layer. The calculated difference 𝑦𝑖 −𝑡𝑖between the actual outputs of the neural network and the target outputs isused as an input to the backward mode to modify the weight of the network. In
40the backward mode, the signal propagates from the output layer to the input layer.
The difference between the actual output of the neural network and the target
output is given to the neural network shown in Fig.4.3.3, which are inputs to thebackward mode. The 𝑦𝑠11 , 𝑦𝑠21 and 𝑦𝑠31 are the outputs of neurons in thehidden layer calculated in the forward mode. The 𝑧 represents the outputs ofeach neurons in the backward mode as shown in Fig. 4.3.4. And we can calculate
it using Eq. (4.3.5).

=1=1

Fig. 4.3.3. Backward mode of error back propagation method.

(y)zFig. 4.3.4. Neuron model in backward mode.
41𝑧 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑 ′ (𝑠) ∑𝑁
𝑛=1 𝑤𝑛 𝑍𝑛
Here, the {𝑍1 , 𝑍2 ,and {𝑤1 , 𝑤2 ,(4.3.5)⋯ 𝑍𝑁 } represent the inputs that are provided to the neuron,⋯ 𝑤𝑁 } are the weights corresponding to the inputs. Then we use

the derived function of the sigmoid function to calculate the outputs of neurons inbackward mode. We use the calculated output 𝑧 of each neuron to make minor
modifications to the weights by Eq. (4.3.6).
𝑤𝑖𝑗−(𝑙+1) = 𝑤𝑖𝑗−(𝑙+1) − 𝜇𝑦𝑠𝑖𝑙 𝑧𝑗(𝑙+1)(4.3.6)

Here, μ is a small positive number that determines the amount of weight change
in each learning process. The 𝑙 represents the number of layers in which theneurons are located, and 𝑖 and 𝑗 represent the 𝑖 -th neurons and the 𝑗 -th
neurons in the layer. The 𝑤𝑖𝑗−(𝑙+1) represents the weight connecting the neuronof number 𝑖 in layer 𝑙 and the neuron of number 𝑗 in layer(𝑙 + 1). Here we use
a three-input-one-output feedforward neural network to explain the error
backpropagation method. But in practice, we may encounter more complicatedsituations, and there may be dozens to hundreds of inputs. So we need to design
the neural network according to the actual situation.
4.4 Design of Neural Network
The reason why neural networks are widely used is because of their learning
ability, which is also considered to be its greatest feature. From the training data
by using a learning method, we could get an approximate relationship that
represents the relationship between the experimental data used as the training
data. Then, use the obtained approximate relationship to give correct output for
inputs other than training data. By optimizing the parameters of the neuralnetwork, a highly accurate approximation can be made [86-87]. In this research,42

learning is performed using the above-mentioned error backpropagation method.
In this study, we decided to use a feedforward neural network consisting of an
input layer, a hidden layer, and an output layer for data processing. In order to beable to get the output we expect, we must properly design the neural network
according to our experimental data. Before designing a neural network, we needto determine the inputs and outputs that are provided to the neural network. Oncethe inputs and outputs are determined, the number of neurons in the input andoutput layers of the neural network is also determined. In the study, the rotation
angle of the sensor is the value we require, so it is used as the output of theneural network. The measured values of the polarization state corresponding to
the angle of rotation of the sensor are provided as an input to the neural network.
Since we are using a three-layer feedforward neural network, after the input
layer and the output layer are determined, in order to obtain a satisfactorylearning result, the appropriate number of neurons in the middle layer should bedetermined. If the number of neurons in the hidden layer is too small, although it
may take a small amount of time to complete the study, it is often difficult to obtainsatisfactory learning results. Conversely, if there are too many neurons in thehidden layer, not only will it take a lot of time to complete the study, but there may
also be overfitting. In both cases, good learning outcomes are not guaranteed. Inorder to determine the appropriate number of neurons for the hidden layer andobtain satisfactory learning results, the Trial & Error method is used [31,82].
The steps of the Trial & Error method are as follows. First, we set up a small
number of neurons for the hidden layer and start learning. If there is a large error
in the learning result, we will increase the number of hidden layer neurons little
43

by little and observe the learning results. If the error becomes smaller, we thencontinue to increase the number of hidden layer neurons. If the error becomeslarger, the number of hidden layer neurons should be reduced. Often this process
has to go through many times until the error is not smaller any longer, and thenumber of neurons at this time is taken as the number of neurons in the hidden
layer.
Similarly, the determination of the number of times of learning is equallyimportant for achieving satisfactory learning outcomes. As with the method ofdetermining the number of hidden layer neurons, we also adjust the number oflearning times while observing the learning results. Similarly, the adjustment of
the weight and the value of the gain during the learning process are also
determined using the Trial & Error method. The gain is a parameter in the sigmoid
function as described previously.
The learning procedure of the error backpropagation method could be
expressed as Fig. 4.4.1 [88]. The learning process is consistent with our previous
description. First, set the initial value for the parameters. The parameters include
the number of neurons in the hidden layer, the number of learning times, thevalues of gain of the sigmoid function, the degree of weight correction and theinitial value of weights. Then, the training data for determining the weights isprovided, the inputs are (𝑥0 , 𝑥1 , … 𝑥𝐿 ), and the outputs are (𝑦0 , 𝑦1 , … 𝑦𝐿 ). After
providing input to the neural network, the output of each neuron in each layer iscalculated to obtain the actual output of the neural network. The program will thenexecute the backward mode to modify the weight value until the set number oflearning times is completed.
44Set initial values for parameters
Whether the convergencecondition is satisfied

YES

End

NO

Set l = 1

Present training data (ｘ𝑙 , ｙ𝑙 )Input to network
Calculate hidden layer neuron
value

Learningloop

Forwardmode

Calculate output layer
neuron value

Loop
for
training
data

Error evaluation
Calculate output layer neuron
value in backpropagation
Calculate hidden layer neuron
value in backpropagation

Backward
mode

Modify weights
YES

If l=LNOll+1

Fig.4.4.1. Learning procedure of the backpropagation method.

45

5. Experiments andResults46

5.1 Verification of twisting when rotating the sensor
In this study, we use a common single-mode fiber that was wound into a circle
shape as a sensor and the sensor is rotated by pulling the Invar wire attached tothe sensor. The state of polarization is measured using a polarization analyzer.
Since we do not know whether the twisting motion affects the experimental results
when the sensor is rotated, it is necessary to detect whether the twisting motionof the rotated sensor affects the experiment before performing the strain
experiment. If there is a twisting effect, then it must be taken into account. The
experimental system for verification of twisting is shown in Figure 5.1.1 [32].

Light
SourcePolarization
Analyzer(a) Measurement of the light source.

SensorLight
SourcePolarization
Analyzer(b) Measurement of light transmitted through the sensor.
Fig. 5.1.1. Experimental systems for verifying twisting.

First, we connected the polarization analyzer directly to the light source tomeasure the state of polarization of the output light of the light source as shownin Fig 5.1.1.a. The next step is shown in Fig.5.1.1.b [32]. We used two 1-m-longfibers and wound them into a circle shape for use as the sensor. Then, we
connected each fiber separately to the light source and conducted the experiment.
In the experiment, we rotated the sensor from 0° to 180° at 10° intervals and47

measured the polarization state of the transmitted light of the sensor. The two
types of fibers used in the experiments were (a) an optical fiber without an outer
jacket and (b) an optical fiber with an outer jacket, and the two types of fibers
have different radii. When the fibers are wound into a circular shape as a sensor,
their bending radii are 0.0215 m and 0.0205 m, respectively [32]. The phasedifference of the sensor was calculated by Eq. 3.3.1 and Eq. 3.3.2. We used themeasured state of polarization of the output light of the light source to calculate
the state of polarization of the transmitted light from the rotated sensor. The
calculation formula used is shown in Eq. 3.2.5. Then, we compared the calculated
and measured values of the state of polarization of the transmitted light from thesensor, and their comparison is shown in Fig. 5.1.2. Here, S1, S2, and S3 denote
the Stokes parameters that represent the state of polarization, which can bedirectly measured by the polarization analyzer in the experiment.
If the twisting motion does not affect the experimental results when the sensor
is rotated, the measured values of the state of polarization of the transmitted lightfrom the sensor should be the same as the calculated values. However, as shownin Fig. 5.1.2, their values do not match with each other no matter which type of
optical fiber is used [32]. This means that the twisting motion has influences on
experiment results when the sensor is rotated. Then, we must also take thistwisting effect into account for calculation when the strain experiment isperformed.

48

Stokes parameters

S1

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

30
60
90
120
150
Rotation angle of sensor (º)

180

Stokes parameters

Measured values of S1
Calculated values of S1 with twist coefficient
Calculated values of S1 without twist coefficient

S2

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

30

60
90
120
150
Rotation angle of sensor (º)

180

Stokes parameters

Measured values of S2
Calculated values of S2 with twist coefficient
Calculated values of S2 without twist coefficient

S3

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

30

60
90
120
150
Rotation angle of sensor (º)

180

Measured values of S3
Calculated values of S3 with twist coefficient
Calculated values of S3 without twist coefficient(a) Results using fibers without an outer jacket.
49

Stokes parameters

S1

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

30
60
90
120
150
Rotation angle of sensor (º)

180

Stokes parameters

Measured values of S1
Calculated values of S1 with twist coefficient
Calculated values of S1 without twist coefficient

S2

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

30

60
90
120
150
Rotation angle of sensor (º)

180

Stokes parameters

Measured values of S2
Calculated values of S2 with twist coefficient
Calculated values of S2 without twist coefficient

S3

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

30

60
90
120
150
Rotation angle of sensor (º)

180

Measured values of S3
Calculated values of S3 with twist coefficient
Calculated values of S3 without twist coefficient(b) Results using fibers with an outer jacket.
Fig. 5.1.2. Experiment for verifying twisting.

50

Furthermore, according to previous research, when the optical fibers arerotated, twisting occurs and the corresponding constant is known as the twisting
coefficient [78]. Then, the state of polarization of the transmitted light from thesensor was calculated from the measured values of the state of polarization of
the light source using Eq. 3.2.5 with twisting coefficient [32]. The results
calculated with consideration of this twisting coefficient are shown in Fig. 5.1.2[32]. If this twisting coefficient can represent the twisting effect generated by therotation of the optical fiber, the results calculated using this twisting coefficient
should be consistent with the measured results shown in Fig. 5.1.2. However, thecalculated values of the state of polarization of the transmitted light from thesensor and the measured values of the state of polarization of the transmitted
light from the sensor still did not match. This means that we can neither ignore
the twisting effect caused by the rotation of the optical fiber nor simply use thetwisting coefficient to calculate the twisting. Therefore, due to the occurrence ofthe twisting and its effect on the state of polarization, we believe that it is possible
to calculate the twisting as a rotated wave plate. The feasibility is also verified by
the following experimental results.
5.2 Experiment using Jones calculus and Results
5.2.1 Experiment system
The strain measurement experimental system used in this study is shown in
Fig. 5.2.1 [32]. We used an LD light source and a polarization analyzer tomeasure changes in the state of polarization of light propagating within the opticalfiber as the sensor rotates. In the experiment, there is an invar line connected tothe sensor, and we pulled the Invar line to rotate the sensor.

51

SensorPolarization
Controller
Light
SourceP

Q

Circulator
SMF
Mirror1

Mirror2

H

Surplus length

Polarization
Analyzer
Fig. 5.2.1. Schematic diagram of the experimental system.
The pulsed light from the light source passes through the polarization controller
and entered the sensor via the optical circulator. The Fresnel reflected light thatwas generated by the reflecting mirrors was incident on the polarization analyzer
through the optical circulator. As can be seen from Fig. 5.2.1, there are two
reflecting mirrors in the experimental system, the reflecting Mirror 1 and thereflecting Mirror 2. Their reflected light is measured by the polarization analyzer.
As long as the state of polarization of the output light of the light source does notchange, the polarization state of the reflected light of the Mirror 1 will not change.
And, the polarization state of the incident light entering the sensor is calculatedusing the polarization state of the reflected light of Mirror 1. On the other hand,
the reflected light from the Mirror 2 is affected by the rotation of the sensor. Thatis, the polarization state of the reflected light of the Mirror 2 changes as the sensor
rotates. In the experiment, since the polarization state of the reflected light of the
Mirror 2 changes with the rotation of the sensor, we use the measured
polarization state to calculate the rotation angle of the sensor. And then using the52rotation angle of the sensor to calculate the displacement of the Invar line andthe strains applied in the experiment. In the experiment, we used pulsed light toperform the measurement, so that when the pulsed light output from the lightsource is reflected by the Mirror 1 and Mirror 2, there will be a time delay between
the two reflected light from Mirror 1 and Mirror 2. We used a 600-m- long opticalfiber to prevent the overlap of the two reflected pulses. The wavelength of thelight source was 1550 nm, the repetition frequency was 1 kHz, the pulse width of
the incident light was set to 3 μs. The maximum sampling frequency of thepolarization analyzer was 1,046 kHz. In this study, we used the polarization statecalculated by the Jones matrix instead of the measured polarization state tocalculate the rotation angle of the sensor. The calculation results given in thefollowing sections also demonstrated the feasibility of the proposed calculation
method.
5.2.2 Experiment and results
In the previous chapter, we described that we propose to use the Jones
matrices of two rotated wave plates to calculate an arbitrary optical path. In orderto get the correct calculation results, we must determine the Jones matrices of
the two rotated wave plates representing each optical path. Therefore, before
performing the strain measurement experiment, we must conduct experiments todetermine the Jones matrices for each optical path, including the optical circulator
part and the surplus length part. When the sensor is rotated, we can use theobtained Jones matrices representing the optical path to calculate the state ofpolarization of the transmitted light from the sensor. This means that we could get
the polarization state of the incident light entering the second sensor during multi-

53

point measurement using the calculus. Then, we measured the polarization stateof the reflected light of the Mirror 1 and Mirror 2 when the sensor is rotated. The
rotation angle of the sensor was obtained from the polarization state of thereflected light from Mirror 1 and Mirror 2, and then the strains applied in theexperiment were determined.
5.2.2.1 Determination of the Jones matrices for the optical circulator.
We use the experimental systems shown in Fig. 5.2.2 to obtain the Jones
matrices of the two rotated wave plates representing the optical circulator section[32]. Here, we assume that the two Jones matrices representing the opticalcirculator section are 𝐉𝟏 and 𝐉𝟐 , respectively [32]. The polarization state oftransmitted light and the polarization state of the reflected light from the opticalcirculator were measured separately.

Polarization
Controller
Light
SourceP

Q

Circulator
Polarization
AnalyzerH(a) Measurement of transmitted light from the optical circulator.

Polarization
Controller
Light
SourceP

Q

Circulator
Mirror 1

H
Polarization
Analyzer(b) Measurement of reflected light from the optical circulator.
Fig. 5.2.2. The experiment of the optical circulator.
The Jones matrices representing the optical circulator were obtained through54

this measurement. In the experiment, first, the linear polarizer (P) of thepolarization controller was set to 0°, and the quarter-wave plate (Q) of thepolarization controller was set to 45°. Then, we measured the polarization stateof the transmitted light and the reflected light from the optical circulator by rotating
the half wave plate (H) from 0° to 90° at 10° intervals. Then, we measured thepolarization state of the transmitted and reflected light from the optical circulator
again by rotating the half-wave plate (H) from 0° to 90° at 10° intervals when thelinear polarizer (P) is set to 0° the quarter-wave plate (Q) is set to 90°.
Initially, 𝐉𝟏 and 𝐉𝟐 can be expressed as the inverse matrices of the Jones
matrices as shown in Eq. (3.2.8) [32]. That is because we use Eq. (3.2.8) to
calculate the polarization state of the reflected light from Mirror 1 using thepolarization state of the transmitted light of the optical circulator. Here, thereflected light from Mirror 1 means the reflected light from the optical circulator.
We obtained the parameters of 𝐉𝟏 (𝜃1 , 𝜙1 ) and 𝐉𝟐 (𝜃2 , 𝜙2 ). The equation for
calculating the polarization state of transmitted light from the optical circulator
using the measured polarization state of the reflected light is expressed as [32]: transmitted light  reflected light from circulator   J 2  J1   from Mirror 1 .


(5.2.1)

Then, we calculated the polarization state of the transmitted light throughchanging the orientation angles (𝜃1 , 𝜃2 ) and phase differences (𝜙1 , 𝜙2 ) of 𝐉𝟏(𝜃1 , 𝜙1 ) and 𝐉𝟐 (𝜃2 , 𝜙2 ) from 0º to 180º at 1º intervals. Then we compared thecalculated and measured values of the state of polarization of the transmitted

55

light to determine the parameter values that minimize the error between them.

Comparison of S1, S2, S3
1

0.8
Stokes parameters

0.6
0.4
0.2

0
-0.2
-0.4
-0.6

-0.8
-1
0

10

20

30

40

50

60

70

80

90

Rotation angle of HWP (º)
Measured values of S1
Calculated values of S1

Measured values of S2
Calculated values of S2

Measured values of S3
Calculated values of S3

Fig.5.2.3. Comparison of the measured values and the calculated values of thetransmitted light from the optical circulator when the quarter-wave plate is at 45º.

56Comparison of S1, S2, S3

1
0.8

Stokes parameters

0.6
0.4
0.2

0
-0.2
-0.4

-0.6
-0.8
-1
0

10

20

30

40

50

60

70

80

90

Rotation angle of HWP (º)
Measured values of S1

Measured values of S2

Measured values of S3

Calculated values of S1

Calculated values of S2

Calculated values of S3

Fig.5.2.4. Comparison of the measured values and the calculated values of thetransmitted light from the optical circulator when the quarter-wave plate is at 90º.

Table 5.2.1 Calculated parameters for the optical circulator.

J1 (𝜃1 , 𝜙1 )J2 (𝜃2 , 𝜙2 )Orientation angle(º)
𝜃1 = 1𝜃2 = 38

Phase difference(º)
𝜙1 = 150
𝜙2 = 106

The comparison between the calculated values of the polarization state of thetransmitted light and the measured polarization state of the transmitted light isshown in Fig. 5.2.3 and Fig. 5.2.4 when the quarter-wave plate is at 45° and 90°,
respectively [32]. Table 5.2.1 lists each of the parameters we obtained [32].
Furthermore, from Fig. 5.2.3 and Fig. 5.2.4, we find that the calculated andmeasured values of the polarization state of the transmitted light of the optical57

circulator are in good agreement with each other. Therefore, by substituting themeasured values of the polarization state of the reflected light from the opticalcirculator and the matrices 𝐉𝟏 (𝜃1 , 𝜙1 ) and 𝐉𝟐 (𝜃2 , 𝜙2 ) obtained above into Eq.
(5.2.1), we can calculate the polarization state of the transmitted light from theoptical circulator. In order to ensure that no interference affects the optical path
during the experiment, we set P, Q, and H to 0°, and measured the polarization
state before and after the experiment. Since the two sets of measurement data
were consistent with each other, it is proved that no interference affects the
optical path.
5.2.2.2 Determination of the Jones matrices for the surplus length.
The surplus length is a length of single-mode fiber connecting the Mirror 1 andthe sensor, as shown in Fig. 5.2.1. Here we used the polarization state of thetransmitted light from the Mirror 1 measured in 5.2.2.1. We assume that the Jones
matrices representing the surplus length are 𝐉𝟑 and 𝐉𝟒 . The state of polarizationof the light that was transmitted through the sensor (without rotating the sensor)
was measured using the experimental system illustrated in Fig. 5.2.5 [32].

SensorPolarization Circulator
SMF
Controller
Mirror1
Light
SourceP

Q

Polarization
Analyzer

H

Fig. 5.2.5. Measurement of the transmitted light from the sensor.
In the experiment, we set the polarization controller’s linear polarizer (P) to 0ºand the quarter-wave plate (Q) to 45º. Then, we measured the polarization stateof the transmitted light from the sensor at 10º intervals when rotating the half58

wave plate (H) from 0º to 90º. Then the Q was set to 90º, and we measured thepolarization state of the transmitted light from the sensor at 10º intervals whenrotating the half-wave plate (H) from 0º to 90º. As described previously, 𝐉𝟑 and𝐉𝟒 are the two rotated wave plates that represent the surplus length, and we can
express them using the Eq. (3.2.8). The parameters of 𝐉𝟑 (𝜃3 , 𝜙3 ) and 𝐉𝟒 (𝜃4 , 𝜙4 )were determined. Furthermore, we can also express the sensor after 𝐉𝟒 as arotated wave plate. We set the orientation angle of the sensor to 0° and the phasedifference to 34.43° to determine the Jones matrices of the surplus length part.
And, the phase difference used here was obtained in Sect. 3.3. Then, we used
the polarization state of the transmitted light from the optical circulator todetermine the polarization state of the transmitted light from the sensor. The
equation we used to perform this calculation is expressed as follows [32]: transmitted light  transmitted light from sensor  sensor   J 4  J 3  from circulator (5.2.2)

Then, by changing the orientation angles (𝜃3 , 𝜃4 ) and phase differences (𝜙3 , 𝜙4 )of 𝐉𝟑 (𝜃3 , 𝜙3 ) and 𝐉𝟒 (𝜃4 , 𝜙4 ) from 0º to 180º at 1º intervals, we calculated thepolarization state of the transmitted light from the sensor. In order to find theparameter values that minimize the difference between them, we compared thecalculated values and measured values of the polarization state of the transmitted
light from the sensor.

59Comparison of S1, S2, S3
1

0.8
Stokes parameters

0.6
0.4
0.2

0
-0.2
-0.4
-0.6

-0.8
-1
0

10

20

30

40

50

60

70

80

90

Rotation angle of HWP (º)
Measured values of S1
Calculated values of S1

Measured values of S2
Calculated values of S2

Measured values of S3
Calculated values of S3

Fig.

5.2.6. Comparison of the measured values and the calculated values of thetransmitted light from the sensor when the quarter-wave plate is at 45º.

Comparison of S1, S2, S3

1
0.8

Stokes parameters

0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1

0

10

20

30

40

50

60

70

80

90

Rotation angle of HWP (º)
Measured values of S1
Calculated values of S1

Measured values of S2
Calculated values of S2

Measured values of S3
Calculated values of S3

Fig. 5.2.7. Comparison of the measured values and the calculated values of thetransmitted light from the sensor when the quarter-wave plate is at 90º.
60Figure 5.2.6 and Figure 5.2.7 show the comparison between the calculated
polarization state of the transmitted light and the measured polarization state of
the transmitted light form the sensor when the quarter-wave plate is at 45º and
90º, respectively [32]. Table 5.2.2 lists each parameter that we obtained [32]. It
can be seen from Fig. 5.2.6 and Fig. 5.2.7 that the calculated and measured
values of the polarization state of the transmitted light from the sensor are very
consistent with each other whether Q is 45º or 90º. This also shows that wecorrectly obtained the Jones matrices representing the surplus length part.
Table 5.2.2 Calculated parameters for the surplus length.

J3 (𝜃3 , 𝜙3 )J4 (𝜃4 , 𝜙4 )Orientation angle(º)
𝜃3 = 25
𝜃4 = 85

Phase difference(º)
𝜙3 = 124
𝜙4 = 139

In order to ensure that no disturbance affects the optical path during theexperiment, we set P, Q, and H to 0°, and measured the polarization state before
and after the experiment. The two sets of measurement data were consistent witheach other, demonstrating that no interference affected the optical path.
5.2.3 Changes in the polarization state due to the rotation of the sensor.
Here, we used the Jones matrices representing the optical circulator and thesurplus length that were obtained in the previous sections to calculate thepolarization state of the transmitted light from the sensor. The measured valuesof the polarization state of the reflected light from Mirror 1 used here are thosethat were measured in Sect. 5.2.2.1 when P, Q, and H were all set to 0º. We
measured the polarization state of the transmitted light from the sensor using theexperimental system shown in Fig. 5.2.5. In the experiment, we set P, Q, and H
to 0° and rotated the sensor from 0° to 180° at 10° intervals. It has been61

demonstrated in Section 5.1 that twisting occurs when the sensor rotated.
Therefore, when calculating the polarization state of the transmitted light from thesensor, we must add a matrix representing the twisting to the calculation equation.
The matrix of the twisting is expressed by Eq. (3.2.8). Thus, the equation for
calculating the state of polarization of the transmitted light from the sensor isexpressed as [32]:

Stokes parameters of transmitted
light from sensor

 transmitted light  reflected light from sensor  sensor  twisting   J 4  J 3  J 2  J1  from Mirror 1 (5.2.3)Comparison of S1, S2, S3

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

20

Measured values of S1
Calculated values of S1

40

60
80
100 120
Rotation angle of sensor (º)
Measured values of S2
Calculated values of S2

140

160

180

Measured values of S3
Calculated values of S3

Fig. 5.2.8. Comparison of the measured values and the calculated values of thetransmitted light from the rotated sensor.
The comparison of the measured values and calculated values of thepolarization state of the transmitted light from the rotated sensor is shown in Fig.
5.2.8 [32]. The calculated values used here were obtained using Eq. (5.2.3). As
can be seen from Fig. 5.2.8, the calculated values of the polarization state of thetransmitted light from the rotated sensor are in good agreement with the62measured values. Therefore, we can use the calculated values to replace themeasured values of the polarization state of the transmitted light of the rotatedsensor in the experiment. Thus, even if another sensor is connected in series
from the back, the polarization state of the light into the second sensor can be
accurately determined. In this way, it is thought that the influence of the front-end
sensor on the rear-end sensor can be solved in the multi-point strain
measurement experiment.
Then, we performed a strain measurement experiment with a single sensor
using the experimental system shown in Fig. 5.2.1. In the experiment, wemeasured the polarization state of the reflected light from Mirror 1 and Mirror 2
when the sensor was rotated. As the sensor rotates, the change in thepolarization state of the reflected light from the sensor is shown in Fig. 5.2.9 [32].
Here, the reflected light of the sensor means the reflected light from Mirror 2.

Comparison of S1, S2, S3

Stokes parameters of reflected lightfrom sensor

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

20

Measured values of S1
Calculated values of S1

40

60
80
100 120
Rotation angle of sensor (º)
Measured values of S2
Calculated values of S2

140

160

180

Measured values of S3
Calculated values of S3

Fig. 5.2.9. The relationship between the polarization state of the reflected lightand the rotation angles of the sensor.

63

As can be seen from Fig. 5.2.9, when the sensor rotates, the calculated valuesof the polarization state of the reflected light from the sensor agrees well with themeasured values as the sensor rotates. This calculated value was obtained using
the measured values of the polarization state of the reflected light from Mirror 1
as described above. Then, a function fitting was performed using the calculated
values shown in Fig. 5.2.9 instead of the measured values. Through function
fitting, we obtained three sets of approximate expressions for the Stokes
parameters expressed by the rotation angle of the sensor. Then we run the C
program to change the rotation angle of the sensor from 0° to 180° at 1° interval,and obtained 181 sets of Stokes parameters [32]. Then, we compared themeasured values of the polarization state of the reflected light from the sensor
with the 181 sets of Stokes parameters to obtain the rotation angle when the error
is minimal. The comparison between the angle obtained in the program and therotation angle applied in the experiment is shown in Fig. 5.2.10 [32].

64Calculated angle of sensor (∘)

180
160

140
120
100
80
0~170°

60

R² = 0.9914

30°~170° R² = 0.9992

40

0~170°

20

30°~170°

0
0

20

40

60

80

100

120

140

160

180

Rotation angle of sensor applied in experiment (∘)
Fig. 5.2.10. Comparison of the calculated angles with the angles that wereapplied in the experiment.
We performed linear approximation separately at intervals of 0º~170º and30º~170º and obtained the values of 𝑅 2 , which are 0.9914 and 0.9992,
respectively. The value of 𝑅 2 is larger in 30º~170º; therefore, the accuracy in thisinterval is higher. Hence, this interval was determined to be the measurement
range. The reason why the error occurs in the interval of 0°~30° is that the change
in the polarization state is very small in the interval from 0°~30° as shown in
Fig.5.2.9. It is believed that the overall error may be reduced in the range of
0°~180° by varying the incident light. Then the strains applied in the experimentwere obtained from rotation angles using a simple mathematical calculation using
Eq. (3.3.3) and (3.3.4). The comparison between the calculated strains and thestrains applied in the experiment is shown in Fig. 5.2.11 [32].

65Calculated strain (×103 με)

200
180
160
140
120
100
80
60
40
20
20

40 60 80 100 120 140 160 180 200
Strain applied in experiment (×103 με)

Fig.5.2.11. Comparison of the calculated strains with the strains that were appliedin the experiment.
As can be seen from Fig. 5.2.11, there is an excellent linear relationshipbetween the calculated strains and the strains applied in the experiment. Table
5.2.3 shows the relationship between the strains applied in the experiment andthe calculated strains [32]. We found that the relative errors between them arewithin 4%. This demonstrates the feasibility of the proposed polarization strain
sensor system using the Jones matrices of two rotated wave plates to calculate
an arbitrary optical path. In addition, we performed the quantification of the strain
measurement range with high accuracy by comparing the calculated strains withthe strains applied in the experiment. Here, the strain measurement range wasquantified by using the calculated values of the polarization state of the reflected
light from the sensor. Therefore, in the strain measurement experiment, we canuse the calculated values to effectively replace the measured values and obtainhigh-precision results. In addition, when multiple sensors are connected in series

66for multi-point strain measurement, measuring the polarization state of reflected
light from each sensor is very time-consuming. This replacement not only saves
a lot of time and labor but also improves the operability of the measurement
system.
Table 5.2.3 Relative errors between the applied and calculated strains.
Rotation angleof sensor(º)

Strain applied in
the experiment30
40

Calculated strain(× 103 με)

Relative error
(%)

34.03
45.38

35.17
44.24

3.3
2.5

50
60
70
80

56.72
68.07
79.41
90.76

58.99
69.20
79.41
91.89

4.0
1.7
0.0
1.3

90
100
110
120
130
140
150
160
170

102.10113.45124.79136.14147.48158.83
170.17181.51
192.86100.97112.31121.39132.73144.08154.29165.63178.11
188.32

1.1
1.0
2.7
2.5
2.3
2.9
2.7
1.9
2.4

(× 103 με)

5.3 Experiment using a Neural Network and Results
In the experiment, we rotated the sensor by pulling the invar wire connected tothe sensor from 0 to 180º at 10º intervals, while the linear polarizer (P), thequarter-wave plate (Q), and the half-wave plate (H) were all set to 0º. Thechanges in the polarization states of the reflected light from the sensor with therotation angle of the sensor are shown in Fig. 5.3.1. As described previously, the67

Stokes parameters S1, S2, and S3 represent the polarization states, which canbe directly measured by the polarization analyzer.

Measured values of the reflected light of sensor
1

Stokes Parameters

0.8
0.6
0.4

0.2
0
-0.2
-0.4

first time of S1
first time of S3
second time of S2
third time of S1

-0.6
-0.8

first time of S2
second time of S1
second time of S3
third time of S2

-1
0

20

40

60

80

100

120

Rotation angle of sensor （°）

140

160

180

Fig. 5.3.1. The relationship between the polarization states and the rotation
angles of the sensor.
In this experiment, we used a neural network to perform data processing todetermine the rotation angle of the sensor. The experimental data shown in Fig.
5.3.1 is used as training data and was provided to the neural network for learning,where the measured polarization state was used as the input and the rotation
angle of the sensor was used as the output. The feedforward neural network weused is illustrated in Fig. 5.3.2. The error back-propagation is the learning method
we used here. Both the time cost and the accuracy of learning results must be
considered when operating the learning. Here, the number of neurons in thehidden layer is set to 10, with the neurons denoted as 𝑛1 , 𝑛2 , ⋯, and 𝑛10 .

68AngleHidden layer

Input layer

Output layer

Fig. 5.3.2. Feed-forward neural network used for learning.

Learning result for training data
Angles ρ outputted by nerural
network (º)

180
150

120
90
60
30
0
0

30

60
90
120
150
Angles ρ applied in experiment (º)

180Fig. 5.3.3. Rotation angles of the sensor applied in the experiment and rotation
angles outputted by the neural network for training data.
In Fig. 5.3.2, 𝑊 denotes the weights, and the process of learning is to adjust
values of weights to make the output of the neural network infinitely approach thetarget output. The measured polarization states were provided to the designed
neural network and the rotation angles applied in the experiment were provided69

to the network as output. During the process of learning, the network is trained tomake the output approach the target output, i.e. the rotation angle of sensor
applied in the experiment. The number of learning times is 1.26 million, the widthof weight correction is 1.0, the gain α is 1.9, and the learning time is 23 seconds.
Fig. 5.3.3 shows the comparison between the angles outputted by the designed
neural network and the angles actually applied in the experiment [31].
As can be seen from Fig. 5.3.3, the angles applied in the experiment and theangles outputted by the neural network agree very well with each other.
Furthermore, the relative errors between them are suppressed to 0.04% or less.
This means that the designed neural network was satisfactorily trained and therelationship between the angles applied to the sensor in the experiment and themeasured polarization states was correctly determined. When the input is given
to the trained neural network, it is expected that the network could yield a correct
output corresponding to that input. As shown in Fig. 5.3.1, we measured threetimes and took three sets of experimental data. In order to test the trained neuralnetwork, we used another set of experimental data as test data and provided thepolarization state as input to the trained neural network to calculate the angles.
The comparison between the calculated angles and the angles that were appliedin the experiment, which were used as test data, is shown in Fig. 5.3.4 [31].

70Learning result for test data
Angles ρ outputted by neuralnetwork (º)

180
150

120
90
60
30
0
0

30

60
90
120
150
Angles ρ applied in experiment (º)

180Fig. 5.3.4. Rotation angles of the sensor applied in the experiment and rotation
angles outputted by the neural network for test data.
As can be seen from 5.3.4, there is a near-linear relationship between theangles applied in the experiment used as test data and the angles outputted by
the trained network. However, an error occurred between the calculated rotation
angle and the actual rotation angle at some angles, when the angle is 20º andthe interval of 110º to 140º. The errors that occurred are not due to the poor
performance of the experimental system, but because we conducted all theexperiments manually. In order to eliminate the error of the experimental data due
to manual operation as much as possible, we obtained the average values of thefirst experimental data and the second experimental data, then we used theaverage values as new training data to perform the learning again. Figure 5.3.5
shows the comparison between the angles outputted by the network and theangles applied in the experiment.

71

Fig. 5.3.5. Rotation angles of the sensor applied in the experiment and rotation
angles outputted by the neural network for new training data.
According to Fig. 5.3.5, the angles applied in the experiment and the angles
outputted by the neural network are in very good accordance with each other.
The relative errors are suppressed to 0.05% or less. This means that the network
was satisfactorily trained and the relationship between the angles applied to thesensor in the experiment and the measured polarization states was correctly
determined. In order to test the trained neural network, we provided the third
measurement data as test data to the network. The polarization state is provided
as input to the network. Figure 5.3.6 shows the comparison between thecalculated angles and the angles that were applied in the experiment, which wereused as test data.

72

Fig. 5.3.6. Rotation angles of the sensor applied in the experiment and rotation
angles outputted by the neural network for new test data.
As can be seen from Fig. 5.3.6, the sensor’s rotation angle of the third
measured polarization state applied in the experiment and the angle obtainedusing the trained neural network are very consistent with each other. Since theangles applied in the experiment and the calculated angles are in good
agreement, we determined the measurement range to be from 0 to 180º. Then,the strains, which is shown in Fig. 5.3.7, were calculated using the rotation angles
in the measurement range with Eq. (3.3.3) and (3.3.4). The measurement rangeof strains that corresponds to the range of rotation angles is calculated to be0~2.0×105 μɛ. Hence, we can obtain the strains accurately with the proposed
measurement method and the data processing method of the neural network.

73

Fig. 5.3.7. Comparison between strains applied in the experiment and calculated
strains.

74

6. Conclusions75

6. Conclusions
In the study, we developed a polarization strain sensor system which is low-costand miniaturized compared with previous polarimetric sensor systems in which
the birefringent or high polarization-maintaining fibers are used. In the sensor
system, the sensor is a commercially available plain single-mode fiber that iswound into a coil shape, and a laser diode light source and a polarization analyzer
are used. In the experiment, the sensor is rotated by pulling the invar wire
connected to the sensor. When the sensor is rotated, changes in the polarization
state of light transmitting in the fiber is measured using the polarization analyzer.
Then, we obtain the rotation angle of the sensor using the movement distance of
the invar wire to calculate the strains that were actually applied in the experiment.
By comparing the calculated strains and the strains applied in the experiment, wedemonstrated that they are in good agreement with each other. The results also
verified that the developed sensor system can be used to measure strains. In
addition, in order to perform multipoint measurement of strains using thedeveloped sensor system, it is necessary to solve the problems that occur inmultipoint measurement. That is, when multiple sensors are connected in series,
the front sensor has an effect on the rear sensor. When the first sensor rotates,
the polarization of the incident light entering the second sensor changes even if
the second sensor does not rotate. This will result in a failure to correctly
determine the rotation angles of the second sensor once both sensors are rotated.
Moreover, once the two sensors are connected in the experiment, the incident
light entering the second sensor cannot be directly measured. To solve thisproblem, we proposed calculating an arbitrary optical path using the Jones

76

matrices of two rotated wave plates. When we obtain two Jones matrices
representing an optical path, we can calculate the polarization state of thetransmitted light of the optical path using the polarization state of the incident lightof the optical path. In the experiment, we used the calculated polarization state
instead of the actually measured polarization state to obtain the strains applied in
the experiment. The measured values of the strains are in good agreement withthe calculated values of the strains, and this demonstrated the feasibility of thismethod using the Jones calculus. In this way, the polarization state of the lightentering the second sensor (the polarization state of the transmitted light from thefirst sensor) can be obtained with the proposed method even if the first sensor
rotates when the multipoint measurement is performed. Thus, multipoint strain
measurements can be achieved by the proposed method using the time delay of
the reflected pulses of a series of sensors.
In this study, we measured the change in the polarization state of the light when
the strains were applied. As mentioned before, the polarization state isrepresented by the Stokes parameters. Thus, a set of strains corresponds tothree sets of Stokes parameters. When multiple sensors are connected in series
for multipoint measurement, the experimental data will become very huge and it
will be very difficult to perform data processing with function fitting. Considering
this problem, we decided to use a three-layer feedforward neural network for dataprocessing. We take the polarization state measured in the experiment as input
and the rotation angle of the sensor as an output to the neural network. The
angles outputted by the designed neural network agree well with the angles
applied in the experiment. In order to verify the trained neural network, we use
77

another set of experimental data as test data and input the polarization state to
the trained neural network to calculate its corresponding rotation angle. And theangles outputted by the trained network are almost the same as the angles usedas the test data. However, errors occur at some angles. The reason for the errors
is considered to be that all experiments were done manually. Then, we performed
learning again using the average values of the two sets of experimental data as
training data. The third set of data is then provided as test data to the trained
network. This time, the rotation angles outputted by the neural network are very
consistent with the rotation angles used as the test data. These results verified
that we can use the developed sensor system and a neural network for strain
measurement. Due to the multi-input and multi-output data processing
capabilities of neural networks, we believe that the multipoint measurement ofstrains can be performed on this basis. As an application of the developed sensor,
it is considered to be useful for measuring landslides. In the experiment, werotated the sensor by pulling the invar wire. Then, if the other end of the invar
wire is connected to a stake fixed on the slope, the invar wire will also be pulledwhen the landslide moves the stake. In this way, it is considered that the extent
of the landslide can be determined from the rotation angle of the sensor.
Therefore, it is believed that the sensor system developed in this study can beused to detect landslides.

78

Acknowledgments
First and foremost, I would like to show my deepest gratitude to my supervisor,
Associate Professor Shinya Sato, who has provided me with valuable guidance,
support, and constant encouragement during the writing of this dissertation as
well as in the research. Secondly, I would like to express my thanks to Professor
Akira Sakai for his guidance during the weekly seminars and support in the
research. Thirdly, I would like to thank Associate Professor Hiroshi Kano for his
advice and support in the research. My sincere appreciation also goes toeveryone in the laboratory for their support in the research. I also owe my thanks
to my friends for their help and advice during the writing of this dissertation. Finally,
my sincere gratitude goes to my beloved family for their loving considerations andgreat confidence in me over the years. This dissertation is dedicated to them all.

79

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