**Training, Education and Research in COVID-19 Times: Innovative Methodological Approaches, Best Practices, and Case Studies**

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**5. Conclusion**

students have a free access to the models through the institutional LAN (the access is also

In the laboratory, they measure the characteristics of the motors and compare them with those from virtual models. They also discuss and explain in detail the motor behavior and the

The work with the virtual models saves time at virtual experimentation, but it requires a careful and detailed explanation and analysis of the obtained characteristics by the teacher-how and why is the behavior of the motor corresponding to the form of the characteristics. Otherwise,

To get feedback from the students and their opinion of introducing the virtual models in the regular teaching, inspired by a questionnaire presented in [9] for Electrical Drives subject we have applied the questionnaire for the Electrical Machines subject and evaluated it (Table 4). Altogether, 36 full-time students of the bachelor study in four groups answered the questions.

**# A B C D E**

Q2 I need the help of instructor to understand graphs 16 13 3 2 2

The student ratings on the evaluation of incorporating the virtual models into teaching – both into lectures and prior laboratory work – are generally positive. It was observed that majority of students found it useful, interesting, and contributing to increasing knowledge about the

The students works with the virtual models also in the following years in the subjects like:

11 18 5 2 0

27 5 2 2 0

29 4 2 1 0

30 5 1 0 0

30 4 2 0 0

available in the student hostels).

virtual models

the laboratory

subjects

continue with their development

**Table 4.** Questionnaire statements and their evaluation

Q5

subject.

influence of motor parameters on the characteristics.

28 E-Learning - Instructional Design, Organizational Strategy and Management

the students do not fully understand the graphs.

Q1 My background knowledge is sufficient to understand working with of the

Q3 Working with models helped me to understand better experimentation in

Q4 I gained new knowledge when working with virtual models (they helped me to understand better the phenomena in electrical machines)

Q6 I would recommend development of similar virtual models also in other

The virtual models of electrical machines are useful and I recommend to

Evaluation criteria: A - strongly agree, B - agree, C - neutral, D - disagree, E - strongly disagree

The aim of this study was to present the methodology and results in the development of virtual models of two types of electrical machines in the MATLAB graphical development environ‐ ment: the one-phase induction motor used in home applications and the stepper motor used widely in automation.

The MATLAB GUIDE, on the one hand, presents a comfortable tool enabling the development of GUI in a very easy and understandable way. On the other hand, it limits the utilization of virtual models by the necessity of the MATLAB program purchase. We do consider it as a limitation because MATLAB is widely distributed and presents a standard tool at the univer‐ sities. Another solution is to use MATLAB server features through the Internet, which enables registered users (students) to utilize the developed interfaces for free. In the boundary case, a stand-alone .exe program can also be developed, but here, some special features of the MATLAB program are lost (Simulink, SymbolicMath toolbox, Animation toolbox, and some other specialties), and they are replaced by bothersome programming.

The two developed GUIs of the one-phase induction motor and of the stepper motor complete the series of virtual modules developed earlier [5] for the subject of electrical machines, like the virtual models of the DC motor, the three-phase AC motors, and the BLDC motor. The virtual models show system performance in various working points and analyze the influence of variable system parameters, modes of supply, and control parameters on system behavior.

The virtual dynamical models, on one hand, contribute to the e-learning support of teaching and, on the other hand, serve for preparation of students before entering the laboratory experimentation. Their utilization also supports and makes more attractive lectures and considerably enhances the explanation of properties of machines.


**2. Temporal biogeographical pattern**

Temporal and biogeographical boundaries cannot be separated from each other. Temporal boundaries eliminate old spatial boundaries and create new ones. Temporal boundaries are usually associated with extinction processes. It is suggested that mass extinctions start at local scales and spill over to higher scales in time [2].


**b. ITIC Curve Scatter Plot**

The ITIC Curve describes an AC input voltage boundary that typically can be tolerated (Sabin, 2000). Events above the upper curve or below the lower curve are presumed to cause the mis-operation of information technology equipment. The curve is not intended to serve as a design specification for products or ac distribution systems. In this case, the number of events, which are below the lower limit of the ITIC curve, in Figure 2, is six, giving a SAFRI-ITIC of six events.

#### **c. SEMI Curve Scatter Plot**

In 1998, the Semiconductor Equipment and Materials International (SEMI) group, power quality and Equipment Ride Through Task force, recommended the SEMI Standard F-47 Curve to predict voltage sag problems for semiconductor manufacturing equipment. Figure 3 shows the application of the data of Table 1on the SEMI curve.

The results obtained from each combination of curves can be presented with the use of tables, such as UNIDEPE DISDIP or ESKOM Voltage sag Table, (Sabin, 2000).

#### **d. Voltage sag co-ordination chart-IEEEStd.493 and 1346**

The chart contains the supply performance for a given site through a given period, and the tolerance of one or more devices. It illustrates the number of events as a function of event severity. Observing the graph shown in Figure 4, there are 5 events per year where the voltage drops below 40% of nominal Voltage for 0.1 s or longer. Equally there are 5 events per year where the voltage drops below 70% magnitude and 250 ms duration.

Fig. 4. Voltage sag co-ordination chart

The advantage of this method is that equipment behavior can be directly compared with system performance, for a wide range of equipment. The disadvantage of the method is that a two-dimensional function is needed to describe the site. For comparison of different sites a smaller number of indices would be preferred.

#### **3.4.2 Calculation methods**

146 Electrical Generation and Distribution Systems and Power Quality Disturbances

The ITIC Curve describes an AC input voltage boundary that typically can be tolerated (Sabin, 2000). Events above the upper curve or below the lower curve are presumed to cause the mis-operation of information technology equipment. The curve is not intended to serve as a design specification for products or ac distribution systems. In this case, the number of events, which are below the lower limit of the ITIC curve, in Figure 2, is six, giving a SAFRI-

> **0.001 0.01 0.1 1 10 100 1000 Duration (seconds)**

In 1998, the Semiconductor Equipment and Materials International (SEMI) group, power quality and Equipment Ride Through Task force, recommended the SEMI Standard F-47 Curve to predict voltage sag problems for semiconductor manufacturing equipment. Figure

> **0.001 0.01 0.1 1 10 100 1000 Duration (seconds)**

The results obtained from each combination of curves can be presented with the use of

The chart contains the supply performance for a given site through a given period, and the tolerance of one or more devices. It illustrates the number of events as a function of event

tables, such as UNIDEPE DISDIP or ESKOM Voltage sag Table, (Sabin, 2000).

**d. Voltage sag co-ordination chart-IEEEStd.493 and 1346**

Total Events: 12 Events Violating SEMI Curve: 5

3 shows the application of the data of Table 1on the SEMI curve.

Total Events: 12

Events Violating ITIC Low er Curve: 6 Events Violating ITIC Upper Curve: 0

**0**

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1**

**Voltage Magnitude (pu)**

**0.5**

**1**

**1.5**

**Voltage Magnitude (pu)**

**2**

**2.5**

**b. ITIC Curve Scatter Plot**

Fig. 2. The ITIC curve scatter plot

**c. SEMI Curve Scatter Plot**

Fig. 3. SEMI Curve scatter plot

ITIC of six events.

#### **a. Method used by Detroit Edison**

The method calculates a "sag score" from the voltage magnitudes in the three phases (Sabin, 2000).

$$S = 1 - \frac{V\_a + V\_b + V\_c}{3} \tag{6}$$

This sag score is equal to the average voltage drop in the three phases. The larger the sag score, the more severe the event is considered to be.

#### **b. Method proposed by Thallam**

A number of site indices can be calculated from the "voltage sag energy" (Thallam, 2000). The "Voltage Sag Energy Index" (VSEI) is the sum of the voltage sag energies for all events measured at a given site during a given period:

$$VSEI = \sum\_{i} \mathbf{E}\_{\text{VS}\_{-i}i} \tag{7}$$

The "Average Voltage Sag Energy Index" (AVSEI) is the average of the voltage sag energies for all events measured at a given site during a given period:

$$AVSEI = \frac{1}{N} \sum\_{i=1}^{N} \mathbf{E}\_{\text{VS\\_}} \tag{8}$$

Power Quality and Voltage Sag Indices in Electrical Power Systems 149

practical. Power quality surveys in the past have just referred to the number of voltage sags per year for a given site. This value could include minor events, which do not affect any

**The Canadian Electrical Association** recommends tracking 4 indices for sag magnitudes (referring to the remaining voltage), of 85%, 70%, 40% and 1%. The latter refers to

**EPRl -Electrotek** suggests the following five magnitudes and three duration ranges to

**a. RMS variation** Frequency for voltage threshold X: with X=90%, 80%, 70%, 50%, 10%: the number of events per year with magnitude below X, and duration between 0.5 cycle

**b. Instantaneous RMS variation** Frequency for voltage threshold X: with X=90%, 80%, 70%, 50%: the number of events per year with magnitude below X, and duration

**c. Momentary RMS variation** Frequency for voltage threshold X: with X=90%, 80%, 70%, 50%: the number of events per year with magnitude below X, and duration between 0.5

**d. Momentary RMS variation** Frequency for voltage threshold I0%: the number of events per year with a magnitude below 10%, and a duration between 0.5 cycle and 3 sec. **e. Temporary RMS variation** Frequency for voltage threshold X: with X=90%, 8070, 70%, 50%, I0%: the number of events per year with magnitude below X, and duration

The duration ranges are based on the definition of instantaneous, momentary and

System Indices are typically a weighted average of the single-site indices obtained for all or a number of sites within the system. The difficulty lies in the determination of the weighting factors. In order to assess any indices for the system, first monitoring of the quality of supply must take place. When the Electric Power Research Institute (EPRI)-Distribution Power Quality (DPQ) program placed monitoring equipment on one hundred feeders, these feeders needed to adequately represent the range of characteristics seen on distribution systems. This required the researchers to use a controlled selection process to ensure that both common and uncommon characteristics of the national distribution systems were well represented in the study sample. Thus a level of randomness is required. Many devices are susceptible to only the magnitude of the variation. Others are susceptible to the combination of magnitude and durationOne consideration in establishing a voltage sag index is that the less expensive a measuring device is, the more likely it will be applied at many locations, more completely representing the voltage quality electricity users are

**ESKOM** (South African Utility), groups voltage sags into five classes (Sabin, 2000):

equipment.

interruptions rather than sags.

characterize voltage thresholds:

between 0.5 cycle and 0.5 sec.

between 3 sec. and 60 sec.

temporary, as specified by IEEE (IEEE Std. 1159, 1995).

and 60 sec.

sec and 3 sec.

**3.5 System indices**

experiencing.

class Y: 80% – 90% magnitude, 20 ms - 3 sec duration class X: 40% - 80% magnitude, 20 ms - l50 ms duration class S: 40% - 80% magnitude, 150 ms - 600 ms duration class T: 0 - 40% magnitude, 20 ms - 600 ms duration class Z: 0 – 80% magnitude, 600 ms - 3 sec duration

A sensitive setting will result in a large number of shallow events (with a low voltage sag energy) and this in a lower value for AVSEI.

The sag event frequency index at a particular location and period is suggested as the number of qualified sag events at a location and period (Thallam & Koellner, 2003).

The System sag count index is the total number of qualified voltage sag events over the number of monitor locations. By the expression qualifying events, it implies a voltage less than 90%, with event duration limited to 15 cycles and energy greater or equal to 100.

#### **3.4.3 Non-rectangular events**

Non-rectangular events are events in which the voltage magnitude varies significantly during the event. A method to include non-rectangular events in the voltage-sag coordination chart is also applicable according to the IEEE defined standard (IEEE Std.493, 1997). Alternatively, the function value can be defined as the number of times per year that the RMS voltage is less than the given magnitude for longer than the given duration.

EPRI-Electrotek mentions that each phase of each ms variation measurement may contain multiple components (Thallam, 2000). Consequently, these phase rectangular voltage sag measurements are easily characterized with respect to magnitude and duration. Approximately 10% of the events are non-rectangular. These events are much more difficult to characterize because no single magnitude-duration pair completely represent the phase measurement.

The method suggested for calculating the indices used by EPRI-Electrotek is called the ''Specified Voltage'' method. This method designates the duration as the period of time that the rms voltage exceeds a specified threshold voltage level used to characterize the disturbance.' The consequence of this method is that an event may have a different duration when being assessed at different voltage thresholds as shown in Figure 5.

Fig. 5. Illustration of "specified voltage" characterization

Most of the single site indices relate the magnitude and duration of the sag and the number of events. These events can be grouped in order to make their counting easier and more practical. Power quality surveys in the past have just referred to the number of voltage sags per year for a given site. This value could include minor events, which do not affect any equipment.

**The Canadian Electrical Association** recommends tracking 4 indices for sag magnitudes (referring to the remaining voltage), of 85%, 70%, 40% and 1%. The latter refers to interruptions rather than sags.

**ESKOM** (South African Utility), groups voltage sags into five classes (Sabin, 2000):

class Y: 80% – 90% magnitude, 20 ms - 3 sec duration

class X: 40% - 80% magnitude, 20 ms - l50 ms duration

class S: 40% - 80% magnitude, 150 ms - 600 ms duration

class T: 0 - 40% magnitude, 20 ms - 600 ms duration

class Z: 0 – 80% magnitude, 600 ms - 3 sec duration

**EPRl -Electrotek** suggests the following five magnitudes and three duration ranges to characterize voltage thresholds:

The duration ranges are based on the definition of instantaneous, momentary and temporary, as specified by IEEE (IEEE Std. 1159, 1995).


*Article* **Information Transfer between Stock Market Sectors: A Comparison between the USA and China**

#### **Peng Yue 1, Yaodong Fan 2,3, Jonathan A. Batten 4 and Wei-Xing Zhou 1,5,6,\***

Received: 13 January 2020; Accepted: 5 February 2020; Published: 7 February 2020

**Abstract:** Information diffusion within financial markets plays a crucial role in the process of price formation and the propagation of sentiment and risk. We perform a comparative analysis of information transfer between industry sectors of the Chinese and the USA stock markets, using daily sector indices for the period from 2000 to 2017. The information flow from one sector to another is measured by the transfer entropy of the daily returns of the two sector indices. We find that the most active sector in information exchange (i.e., the largest total information inflow and outflow) is the *non-bank financial* sector in the Chinese market and the *technology* sector in the USA market. This is consistent with the role of the non-bank sector in corporate financing in China and the impact of technological innovation in the USA. In each market, the most active sector is also the largest information sink that has the largest information inflow (i.e., inflow minus outflow). In contrast, we identify that the main information source is the *bank* sector in the Chinese market and the *energy* sector in the USA market. In the case of China, this is due to the importance of net bank lending as a signal of corporate activity and the role of energy pricing in affecting corporate profitability. There are sectors such as the *real estate* sector that could be an information sink in one market but an information source in the other, showing the complex behavior of different markets. Overall, these findings show that stock markets are more synchronized, or ordered, during periods of turmoil than during periods of stability.

**Keywords:** information transfer; transfer entropy; stock markets; econophysics

#### **1. Introduction**

Complex systems, such as financial markets, are usually composed of many subsystems; in the case of financial markets, information flows and interactions within the market itself are rarely investigated even though they are critical to driving the complex dynamics of the complex system as a whole. Many methods have been proposed to unveil these different relationships among subsystems, such as correlations including simple correlation analysis [1,2], Granger causality [3], nonparametric approaches such as the thermal optimal path method [4–6], and mutual information analysis [7–9]. These different approaches have their own advantages and limitations. Importantly, while Granger causality is commonly used to identify time-varying single or bidirectional causality in economics, it is sensitive to sample period selection and complexity in the underlying time series, as well as having other issues [10,11].

In this paper, we use an alternative approach termed transfer entropy to identify the information transfers between industrial sectors in the world's two largest economies: the USA and China. Transfer entropy, as a kind of log-likelihood ratio [12], is a measure that quantifies information flow based on the probability density function (PDF). Better than correlations or Granger causality, transfer entropy not only identifies the direction of the information flow but also quantifies the flows between different subsystems. In other words, it is capable of quantifying the strength and direction of the interaction between different subsystems at the same time. This approach has found wide application [13–21]. Furthermore, variation and extensions of transfer entropy have been developed that are suitable for different situations [22], such as symbolic transfer entropy [23].

There are many studies adopting the concept of transfer entropy to economic systems such as financial time series [18,24,25], stock market indices [26,27], composite index and the constituent stocks [28,29], and indices of industry sectors of a stock market [30].

Stock price fluctuations reflect both global and local news as well as news within a subsystem. There are also well-known calendar anomalies related to business cycle and market participants sector rotations [31]. In a related work, Oh et al. investigated the information flows among different sectors of the Korean stock market [30]. They measured the amount of information flow and the degree of information flow asymmetry between industry sectors around the subprime crisis and identified the insurance sector as the key information source after the crisis. Although the authors do not attribute a economic basis for this finding, it is likely linked to the insurance sector acting as a leading indicator of risk in the economy. In this work, their analysis is extended and a comparative study is performed on the information transfer among different industry sectors of the Chinese and the USA stock markets. These two markets are respectively the largest emerging and developed stock markets associated with the two largest economies in the world.

The rest of this paper is organized as follows. Section 2 describes the method for calculating symbolic transfer entropy and the sector indices time series for the Chinese and the USA stock markets. Section 3 presents the empirical results about the information flows between stock market sectors and its relationship with market states. Section 4 concludes this work.

#### **2. Method and Data**

#### *2.1. Symbolic Transfer Entropy*

Schreiber was the first to use transfer entropy to measure information transfer and detect asymmetry in the interactions among subsystems [13]. He treated a sleeping human's breath rate time series and heart rate time series as two subsystems and found that the information flow from the heart to the breath signal is dominant. To explore the transfer entropy between two time series, there are various approaches in the literature. We need to briefly summarize what the other approaches are and why the symbolic transfer method is used. We use the symbolic transfer entropy introduced by Staniek and Lehnertz [23]. Consider two different daily closing prices time series {*Xt*} and {*Yt*}, *t* = 1, 2, ... , *L*, which have the same length *L*. Closing prices are used to ensure that prices factor in local market news as well as intramarket news from the various sectors. Transfer entropy *TS y*→*x* assumes that *Xt* is influenced by the previous *l* states of the same variable and by the *m* previous states of variable *Y*, for financial markets, only the day before is important [32]. Hence, we use *l* = *m* = 1 in this study. The procedure to calculate the symbolic transfer entropy *TS y*→*x* from time series {*yt*} to {*xt*} is briefly described in the following five steps:

First, we adopt the log returns {*xt*} instead of the original price time series {*Xt*} by

$$\ln x\_t \equiv \ln(X\_t) - \ln(X\_{t-1}) \tag{1}$$

where *Xt* is the closing price of the index on the *t*th trading day.

Second, the returns are discretized into *q* nonoverlapping windows of equal length Δ. If there are too many windows, the chance of having particular combinations drops very quickly, making the calculation of probabilities slower and less informative [32]. Hence, it is irrational if *q* is too large or too small. Marschinski and Kantz consider *q* = 2 and *q* = 3 in their research [24]; Sandoval uses *q* = 24 and *q* = 6 [32]. We aim at finding the optimal *q* to maximize the transfer entropy difference between two time series meanwhile minimizing the calculation cost. In our comparative investigations, the parameter *q* varies from 2 to 22 with a moving step of 1. We find that when *q* ≥ 15, the difference becomes significantly nonzero. Considering the calculation cost and the strength of transfer entropy, in this work, we use *q* = 15. We obtain the maximum value *x*max and minimum value *x*min of the time series *xt* under investigation. The length of each interval is Δ*x* = [*x*max − *x*min]/*q* and the *k*th interval is [*x*min + (*k* − 1)Δ*x*, *x*min + *k*Δ*x*). Similarly, we repeat the procedure for *yt* and its Δ = Δ*y* is usually different from Δ*x*.

Third, the log return time series *x*ˆ and *y*ˆ are described as

$$\mathfrak{X}\_{l} = f(\mathbf{x}\_{l}) = k\_{\mathbf{x}} \text{ and } \mathfrak{Y}\_{l} = f(y\_{l}) = k\_{\mathbf{y}}, \quad k\_{\mathbf{x}}, k\_{\mathbf{y}} = 1, 2, \cdots, q,\tag{2}$$

where *xt* ∈ [*x*min + (*kx* − 1)Δ*x*, *x*min + *kx*Δ*x*) and *yt* ∈ [*y*min + (*ky* − 1)Δ*y*, *y*min + *ky*Δ*y*).

Fourth, the number of elements in the *q*th interval are denoted by *x*ˆ *q t* and *y*ˆ *q t* , respectively, and then calculate the probabilities *p*(*x*ˆ*t*) = *x*ˆ *q t* /(*L* − 1) and *p*(*y*ˆ*t*) = *y*ˆ *q t* /(*L* − 1) and the joint probabilities *p*(*x*ˆ*t*, *y*ˆ*t*), *p*(*x*ˆ*t*, *x*ˆ*t*+1) and *p*(*x*ˆ*t*+1, *x*ˆ*t*, *y*ˆ*t*).

Fifth, in information theory, different bases of entropy lead to different units of entropy. Base 2 is the most widely applied in transfer entropy for most of empirical works. Therefore, in our study, we use Base 2 to calculate transfer entropy. The symbolic transfer entropy from time series {*yt*} to time series {*xt*} is calculated as

$$T\_{y \rightarrow \mathbf{x}}^{\rm S} = \sum\_{\p\_{t+1}, \p\_{t, \hat{\jmath}\_t} \atop t \atop t} p(\p\_{t+1}, \p\_{t, \hat{\jmath}\_t} \hat{\jmath}\_t) \log\_2 \frac{p(\p\_{t+1} | \p\_{t, \hat{\jmath}\_t} \hat{\jmath}\_t)}{p(\hat{\chi}\_{t+1} | \hat{\chi}\_t)},\tag{3}$$

where the joint probability *p*(*x*ˆ*t*+1, *x*ˆ*t*, *y*ˆ*t*) means the probability that the combination of *x*ˆ*t*+1, *x*ˆ*t* and *y*ˆ*t* occurs, while *p*(*x*ˆ*t*+1|*x*ˆ*t*, *y*ˆ*t*) and *p*(*x*ˆ*t*+1|*x*ˆ*t*) are the conditional probabilities that *x*ˆ*t*+1 has a particular value when the values of previous samples *x*ˆ*t* and *y*ˆ*t* are known and *x*ˆ*t* is known, respectively. Since

$$p(\mathbf{\hat{x}}\_{t+1}|\mathbf{\hat{x}}\_{t},\mathbf{\hat{y}}\_{t}) = \frac{p(\mathbf{\hat{x}}\_{t+1},\mathbf{\hat{x}}\_{t},\mathbf{\hat{y}}\_{t})}{p(\mathbf{\hat{x}}\_{t},\mathbf{\hat{y}}\_{t})} \text{ and } \ p(\mathbf{\hat{x}}\_{t+1}|\mathbf{\hat{x}}\_{t}) = \frac{p(\mathbf{\hat{x}}\_{t+1},\mathbf{\hat{x}}\_{t})}{p(\mathbf{\hat{x}}\_{t})},\tag{4}$$

we can simplify Equation (3) and obtain

$$T\_{y \rightarrow \mathbf{x}}^{\mathbb{S}} = \sum\_{\hat{\mathfrak{x}}\_{l+1}, \hat{\mathfrak{x}}\_{l}, \hat{\mathfrak{y}}\_{l}} p(\mathfrak{x}\_{l+1}, \mathfrak{x}\_{l}, \hat{\mathfrak{y}}\_{l}) \log\_{2} \frac{p(\mathfrak{x}\_{l+1}, \hat{\mathfrak{x}}\_{l}, \hat{\mathfrak{y}}\_{l}) p(\mathfrak{x}\_{l})}{p(\mathfrak{x}\_{l+1}, \hat{\mathfrak{x}}\_{l}) p(\mathfrak{x}\_{l}, \hat{\mathfrak{y}}\_{l})}. \tag{5}$$

This expression is used for the estimation of the symbolic transfer entropy.

#### *2.2. Data Description*

To conduct our analysis, we selected two sets of data from two major stock markets: the Chinese and the USA stock market. The Chinese stock market is the largest emerging market, while the US stock market is the worlds largest developed stock market.

For the Chinese stock market, we retrieved and analyzed the SWS sector indices issued by Shenyin & Wanguo Securities Co., Ltd. (http://www.swsresearch.com). In total, this gave 28 sector indices of the Chinese stock market, and covered 3508 individual stocks. For each sector index series, there were 4359 daily prices from 4 January 2000 to 29 December 2017. The various sectors with their corresponding six-digit codes include: agriculture and forestry (801010), mining (801020), chemical (801030), steel (801040), non-ferrous metals (801050), electronic (801080), household appliances (801110), food and drink (801120), textile and apparel (801130), light manufacturing (801140), biotechnology (801150), utilities (801160), transportation (801170), real estate (801180), commercial trade (801200), leisure and services (801210), composite (801230), building materials (801710), building and decoration (801720), electrical equipment (801730), national defense (801740), computer (801750), media (801760), communications (801770), bank (801780), non-bank financial (801790), automobile (801880), and mechanical equipment (801890).

For the US stock market, we chose 16 sector indices composed by Thompson Reuters Co., Ltd. (http://www.thomsonreuters.cn). For each sector index time series, there were 4695 daily prices from 3 January 2000 to 29 December 2017. The differences in day count in the two series are due to differences in holidays in the two countries. The stock sectors are appliances resources (M3L), banking/investment services (BIL), cyclical construction producers (YPL), cyclical consumer services (CRL), energy (E2L), food/beverages (FBL), healthcare services (HSL), industrial and commercial services (IVL), industrial goods (IGL), mineral resources (MRL), pharmaceuticals/MD research (PHL), real estate (REL), retailers (RTL), technology (TEL), transportation (TRL), and utilities (U2\$). We use the concept of symbolic transfer entropy to detect and measure the information flows among the return time series of the 28 sector indices of the Chinese stock market and the 16 sector indices of the USA stock market.

#### **3. Results and Discussion**

#### *3.1. Symbolic Transfer Entropy and Degree of Asymmetric Information Flow of the Whole Samples*

As mentioned in Section 1, symbolic transfer entropy can proxy for the strength and direction of the information flow between two time series. Following Oh et al. [30], we used the degree of asymmetric information flow to measure the information effect between two stock sectors, which is defined as

$$
\Delta T\_{i \to j}^{S} = T\_{i \to j}^{S} - T\_{j \to i}^{S}. \tag{6}
$$

It follows that Δ*TS j*→*i* = −Δ*TS i*→*j* . We show the calculation results of our datasets in four heat maps (top row for the Chinese sectors and bottom row for the US sectors) of *TS i*,*j* and Δ*TS i*,*j* in Figure 1, in which each cell shows the value of *TS* (left plot) or Δ*TS* (right plot) from sector *i* to sector *j*. We observe that the values in the diagonal matrices *TS i*,*j* and Δ*TS i*,*j* are zeros, which is trivial and can be understood from the concept of symbolic transfer entropy. We find that the *non-bank financial* sector (code 790) has roughly the highest *TS* values for both inflows and outflows among the Chinese sectors, and the *technology* sector (code TEL) has roughly the highest *TS* values for both inflows and outflows among the US sectors; the *non-bank financial* sector comprises three Level 2 sectors in the SWS index system which are *security*, *insurance*, and *multivariate financial*. This suggests that during the sample period from 2000 to 2017, the *non-bank financial* sector and the *technology* sector were respectively the most active in the two stock markets. That is, there was more information exchange between these sectors with the other sectors in their own stock markets than between other sectors in the same stock market.

**Figure 1.** Heat maps of the symbolic transfer entropy matrix *TS i*,*j* (left matrices) and the degree of asymmetric information flow Δ*TS i*,*j* (right matrices) between the 28 Chinese stock market sectors (2000-2017, top matrices) and the 16 USA stock market sectors (2000-2017, bottom matrices). To simplify the label, we use the last three digits of each 6-digit code to represent the corresponding Chinese stock market sector.

#### *3.2. Average Inflow and Outflow*

For each sector *i*, the average outflow *F*out,*i* and inflow *F*in,*i* of information can be calculated as follows [30]:

$$F\_{\text{out},i} = \frac{1}{n-1} \sum\_{p \neq i} T\_{i \to p}^S \tag{7a}$$

and

$$F\_{\rm in,i} = \frac{1}{n-1} \sum\_{p \neq i} T\_{p \to i\prime}^{S} \tag{7b}$$

where the points with *i* = *j* are not included. Figure 2a,c show the bar charts of the average information inflows and outflows for all the sectors. This figure confirms that the *non-bank financial* sector (code 790) and the *technology* sector (code TEL) were the most active sectors in information exchange, respectively. We also find that the more information a sector sends out to other sectors, the more information it receives from others generally. Therefore, the outflow and inflow are positively related to each other. We present in Figure 2b,d the scatter plot of *F*out,*i* against *F*in,*i*, which confirms a significant positive

correlation. The least-squares regression results in the following linear relationship for the Chinese market are as follows:

$$F\_{\rm out,i} = 0.724 F\_{\rm in,i} + 0.037,\tag{8a}$$

where the *p*-values of the two coefficients are respectively 3 × 10−15 and 2 × 10−6 and the adjusted *R*2 is 0.908. Similarly, for the USA market we have

$$F\_{\rm out,i} = 0.291 F\_{\rm in,i} + 0.046\_{\prime} \tag{8b}$$

where the *p*-values of the two coefficients are 6 × 10−4 and 5 × 10−8, respectively, and the adjusted *R*2 is 0.548. It is clear from this simple estimation that the linear relationship is more significant for the Chinese stock market. We argue that the linearity reflects the degree of traders' actions on the idiosyncratic traits of market sectors. The higher linearity of the Chinese stock market implies that the traders in the Chinese market are more irrational, such that their behavior is less reflected in the idiosyncratic traits of market sectors in their decision-making process.

**Figure 2.** Average outflow *F*out,*i* and inflow *F*in,*i* of information for stock market sectors. (**a**) Bar chart for the Chinese stock market. (**b**) Scatter plot for the Chinese stock market. (**c**) Bar chart for the USA stock market. (**d**) Scatter plot for the USA stock market.

We also use the average degree of asymmetric information flow Δ*Fi* to measure the net information of sector *i* being sent to other sectors, which is defined as follows [30]:

$$
\Delta F\_i = F\_{\text{out},i} - F\_{\text{in},i} \,\text{.}\tag{9}
$$

We illustrate in Figure 3 the average degree of asymmetric information flow Δ*F* of the sectors in descending order for the two stock markets.

**Figure 3.** Degree of asymmetric information flow Δ*F* in descending order of industry sectors. (**a**) Chinese stock market sectors. (**b**) USA stock market sectors.

Among all the 28 Chinese sectors, the *bank* sector (code 780) has the highest Δ*F* value, while the Δ*F* value of the *non-bank financial* sector is the lowest. This finding suggests that the *bank* sector has the highest net outflow of information and is thus the most influential sector, while the *non-bank financial* sector is the most influenced sector. If we regard the Chinese stock market as an information transfer system, the *bank* sector is a big information source, influencing other sectors, while the *non-bank financial* sector is a big information sink, influenced by other sectors. Concerning the absolute Δ*F* value, we find that the *biotechnology* (code 150) is the closest one to zero, which indicates that the strength of information outflows and inflows are approximately equal and there is little net information transferred between the *biotechnology* sector and the whole market.

Among all the 16 US sectors, the *energy* sector (code E2L) has the highest Δ*F* value, while the Δ*F* value of the *technology* sector (code TEL) is the lowest. This suggests that the *energy* sector has the highest net outflow of information and is thus the most influential sector, while the *technology* sector is the most influenced sector. Therefore, the *energy* sector is a big information source, influencing other sectors, while the *technology* sector is a big information sink, influenced by other sectors. When we consider the absolute Δ*F* value, we find that the *appliances* sector (code M3L) is the closest one to zero, which indicates that the strength of information outflows and inflows are approximately equal and there is little net information transferred between the *appliances* sector and the whole market.

Although the sectors in both markets are similar, they play different roles in the two information transfer processes. For instance, the *real estate* sector is an information sink in the Chinese market but an information source in the US market. These results highlight the importance of the *real estate* sector in driving economic output in China and its less significant role in the US.

#### *3.3. Yearly Evolution of Symbolic Transfer Entropy and Degree of Asymmetric Information Flow*

Economic sectoral relationships are known to be unstable and change over time. For example, Bernanke (2016) highlighted the changing correlation between the energy and industrial sectors in the US over the last decade. To qualify the evolution of information flows over time, we calculated the symbolic transfer entropy matrix *TS*(*t*) and the asymmetric average information flow Δ*TS*(*t*) for each year *t*. The four *TS*(*t*) heat maps of the Chinese stock market for years 2000, 2003, 2007, and 2011 are illustrated in Figure 4 , and the four *TS*(*t*) heat maps of the US stock market for years 2000, 2003, 2007, and 2011 are illustrated in Figure 5, respectively. For the Chinese stock market, it is found that the heat maps share some pattern of similarity. For instance, some relative bright lines emerge vertically and horizontally, echoing the pattern in Figure 4. However, these heat maps also exhibit remarkable differences. The most significant feature is that the heat maps become brighter over time, indicating that there are more information transfers among different sectors with the development of the stock market. The corresponding four heat maps of the asymmetric information flow Δ*TS*(*t*) are shown

in Figure 6. A similar evolution of patterns is observed in the US stock markets, which is shown in Figures 5 and 7. However, we do not observe a monotonic increase in information flows among the US sectors, in which the information flows among sectors were smaller in 2011.

**Figure 4.** Heat maps of the symbolic transfer entropy matrix *TS i*,*j* for four years (2000, 2003, 2007, and 2011) of the Chinese stock market.

**Figure 5.** Heat maps of the symbolic transfer entropy matrix *TS i*,*j* for four years (2000, 2003, 2007, and 2011) of the USA stock market.

**Figure 6.** Heat maps of the symbolic transfer entropy matrix Δ*TS i*,*j* for four years (2000, 2003, 2007, and 2011) of the Chinese stock market.

**Figure 7.** Heat maps of the symbolic transfer entropy matrix Δ*TS i*,*j* for four years (2000, 2003, 2007, and 2011) of the USA stock market.

To further quantify the evolution of information flows, we calculated the average of the symbolic transfer entropy matrix *TS*(*t*) for each year *t* as follows:

$$\left\langle T^S(t) \right\rangle = \frac{1}{n(n-1)} \sum\_{i \neq j} T^S\_{i \to j}(t) \,. \tag{10a}$$

where the diagonal with *i* = *j* is not included, and the average asymmetric information flow Δ*TS*(*t*) for each year *t* is measured as follows:

$$\left<\Delta T^{\mathbb{S}}(t)\right> = \frac{2}{n(n-1)}\sum\_{i=1}^{n}\sum\_{j=1}^{i} |\Delta T\_{i \to j}^{\mathbb{S}}(t)|\_{\prime} \tag{10b}$$

where the lower triangle (i.e., the part with *i* ≤ *j*) is not included. We note that there are no objective criteria to determine the window size. Too long windows will result in too few data points and vague evolution paths, while too short windows lead to less statistics and more noise [33]. The choice of one year is a trade-off.

The evolutionary trajectories of the average symbolic transfer entropy *TS*(*t*) and the average asymmetric information flow Δ*TS*(*t*) from 2000 to 2017 of the Chinese stock market are presented in Figure 8a,b, respectively, while the results for the US stock market are presented in Figure 8c,d. For the Chinese stock market, we observe two local minima around 2001 and 2016 for *TS*(*t*) and three local minima around 2001, 2008, and 2016 for Δ*TS*(*t*). This observation is of particular interest, because the three periods correspond to key periods of market volatility associated with the market crashes in June 2001 [34], December 2007 [35], June 2009 [35], June 2015 [36], and January 2006 [37]. For the US stock market, we observe four local minima around 2001, 2008, 2011, and 2016 for *TS*(*t*) and three local minima around 2001, 2011, and 2015 for Δ*TS*(*t*), which correspond to the 9/11 terrorist attack in 2001 [38], the subprime mortgage crisis in 2007 [39], the July–August 2011 stock market crash [40], and the 2015–16 stock market selloff beginning in the United States on 18 August 2015. It is documented for other types of networks that the structure of networks usually changes around large market movements (see [41] and the references therein).

**Figure 8.** (**a**) Evolution of the average symbolic transfer entropy *TS*(*t*) from 2000 to 2017 of the Chinese stock market. (**b**) Evolution of the average asymmetric information flow Δ*TS*(*t*) from 2000 to 2017 of the Chinese stock market. (**c**) Evolution of the average symbolic transfer entropy *TS*(*t*) from 2000 to 2017 of the USA stock market. (**d**) Evolution of the average asymmetric information flow Δ*TS*(*t*) from 2000 to 2017 of the USA stock market.

We conclude that, during market turmoil periods, both the average information transfer and the average asymmetric information flow are lower than in stable states. This conclusion is not surprising. During bubbles and antibubbles, investors exhibit stronger convergence in decision making. The majority of investors buy stocks during bubbles and sell stocks during antibubbles. Although stock markets have higher volatility during periods of turmoil, investors' actions are more synchronized. In other words, stock markets are more integrated during periods of turmoil than during periods of stability.

#### **4. Conclusions**

In this work, we compared the information transfer between industry sectors in the Chinese and US stock markets based on their symbolic transfer entropy. We used daily returns of key sector indices from 2000 to 2017. The results of this work offer several important insights into information flows between industry sectors. First, we find that the most active sector in information exchange is the *non-bank financial* sector in the Chinese market and the *technology* sector in the US market. Second, concerning the net information flow of individual sectors, we find that the main information source is the *bank* sector in the Chinese market and the *energy* sector in the US market, while the information sink is the *non-bank financial* in the Chinese market and the *technology* sector in the US market. The two information sinks with the largest net information inflow in the two markets are exactly the two most active sectors with the largest information transfer. Third, the same sector may play different roles in the two markets. For example, the *real estate* sector is an information sink in the Chinese market but an information source in the US market. Thus, the US stock market is expected to react to demand related to news originating from the housing sector, such as building approvals, whereas in China this is not the case since the markets are driven by supply side factors such as changes in bank lending.

We also investigated the evolution of the yearly information transfer for both markets. It is found that the local minima of the average symbolic transfer entropy *TS*(*t*) and the average asymmetric information flow Δ*TS*(*t*) correspond to periods of market turmoil. We argue that stock markets are more integrated during periods of turmoil than in stable periods, which results in smaller entropy.

Note that while there have been several studies that use entropy-based techniques to predict market fluctuations and crashes [42–47] or measures [48–50], in this study we argue that the average symbolic transfer entropy *TS*(*t*) and the average asymmetric information flow Δ*TS*(*t*) do not have a direct predictive power for market crashes. Further research is required to better understand the dynamics of market crashes, which are likely not driven by historical correlations but rather by behavioral factors.

**Author Contributions:** Data curation, P.Y.; Formal analysis, P.Y.; Investigation, P.Y., Y.F., J.A.B. and W.-X.Z.; Methodology, P.Y.; Supervision, W.-X.Z.; Writing—original draft, P.Y.; Writing—review & editing, Y.F., J.A.B. and W.-X.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partly supported by the National Natural Science Foundation of China (U1811462, 71532009), the Shanghai Philosophy and Social Science Fund Project (2017BJB006), the Program of Shanghai Young Top-notch Talent (2018), and the Fundamental Research Funds for the Central Universities.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).


**1. Introduction**

Neuropathic pain is a condition that is caused by damage to the nervous system as a result of physical trauma, chemotherapy, infection, metabolic disease, or autoimmune disorders [1,2]. This chronic pain condition affects 7–10% of the population and, unfortunately, current drug treatments have poor efficacy and tolerability [3,4]. Patients are typically unresponsive to analgesics and opioids. In addition to pain, patients also suffer from sleep disturbances, anxiety and depression, and reduced quality of life [5]. The incidence of neuropathic pain is expected to increase in the coming years due in part to the diabetes epidemic, improved cancer survival rates, and age [6]. Therefore, new approaches to treat pain in these patients are clearly needed.

Cannabis has been used for centuries to treat pain, and the plant contains a number of pharmacologically active compounds including cannabinoids and terpenes that might have anti-nociceptive properties [7,8]. The two most abundant and studied cannabinoids in the plant are Δ9-tetrahydrocannabinol (THC) and cannabidiol (CBD), and the ratio of these two compounds can vary greatly between cultivars and subtypes of *Cannabis*. Both THC and CBD have been shown to reduce neuropathic pain in animal models [9–13]. However, patients rarely take pure THC or CBD for pain. Additionally, THC and CBD are normally administered together at varying ratios depending on the product consumed. Furthermore, the anti-nociceptive effects of these varying ratios of THC and CBD remain underexplored. Instead, studies have largely focused on THC:CBD co-administered at 1:1 ratios, commonly found in the European-approved Sativex (nabiximols) that has been shown to be effective at treating pain in patients with certain conditions [14–17]. We

**Citation:** Sepulveda, D.E.; Vrana, K.E.; Graziane, N.M.; Raup-Konsavage, W.M. Combinations of Cannabidiol and Δ9-Tetrahydrocannabinol in Reducing Chemotherapeutic Induced Neuropathic Pain. *Biomedicines* **2022**, *10*, 2548. https://doi.org/10.3390/ biomedicines10102548

Academic Editor: Giuseppe Tringali

Received: 1 September 2022 Accepted: 10 October 2022 Published: 12 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Hershey, PA 17033, USA

therefore set out to assess how different combinations of THC and CBD, as both pure compounds and from unfractionated plant extracts, affect mechanical hypersensitivity in a mouse model of chemotherapeutic-induced peripheral neuropathy (CIPN).

#### **2. Materials and Methods**


**Abstract**

This chapter aims to address the quantum signal role and properties in optical fiber application mainly in quantum communication. It covers the general discussion on quantum bits and optical waveguiding properties. The highlight of this chapter lies in the discussion of the quantum fictitious force of anti-centrifugal force which was first reported in 2001. Under this condition, the free particle experience an attractive potential towards the rotating center of a bent waveguide structure. A lot of theoretical work has been carried out to observe this quantum phenomenon. However, no intensive experimental work has been carried out to date. With the advancement of nano-fabrication technology and quantum experimental, it provides a bright potential to observe these phenomena. Thus, we proposed a promising material of Lithium Niobate on Insulator to serve as a waveguiding platform to study this quantum effect experimentally. The discussion is extended to perceive the relation between Schrodinger and Helmholtz's equation corresponding to this effect.

**Keywords:** quantum anti-centrifugal force, waveguide, rib waveguide, LNOI, spatial mode

## **1. Introduction**

Optical fiber is known as one of the passive devices that serve as the transmission medium in optical communications. This cylindrical-shaped dielectric waveguide consists of the higher refractive index inner core surrounded with slightly lower index cladding which operates based on total internal reflection phenomena. To date, the advancement in optical fiber technology led to the development of new fiber designs and components to establish multiple co-existing data channels based on light propagation over distinct transverse optical modes [1]. Apart from that, there have been tremendous research and developments in quantum information processing since Richard Feynman's seminal talk on the future of quantum computing 40 years ago. These developments have been motivated by the higher demand for higher transmission bandwidth in telecommunication infrastructure. Thus, optical fiber has played its main role in the success of this technology mainly due to its high transparency and high-bandwidth support [2]. Thus, the discussion of this chapter covers the general idea of quantum bits which has become the backbone in quantum information.

Apart from that, it is found that the bending of optical waveguide has led to the signal distortion which increase the propagation losses. For the pass few years, researcher put an extra effort to observe this phenomena which is known an quantum anti-centrifugal force from a quantum physics point of view.

Therefore, the aim of this chapter lies on the discussion of poorly reviewed quantum anti-centrifugal force. Furthermore, we perceive the correlation of modern interpretation of Schrodinger equation with the classical Helmholtz equation correspond to this quantum fictitious force. The observation of this effect is extended into the real application of Lithium Niobate on Insulator rib waveguide as it found to be an extensive platfrom towards the development of quantum information technology.

## **2. Optical field propagation**

The optical waveguide structure consists of high refractive index core, surrounded by slightly lower refractive index. It operates based on total internal reflection phenomena which allow the electromagnetic wave guiding towards core. The electromagnetic plane waves propagation in homogeneous and isotropic media such as glass and diamond are governed by Maxwell's equation:

$$
\nabla \times \mathbf{E} + \mu \frac{\partial H}{\partial t} = \mathbf{0} \tag{1}
$$

does not hold for all coordinate systems [7]. Therefore, the need to choose for a suitable coordinate system and boundary conditions are crucially needed prior to

> *∂*2 *∂φ*2 þ *k*2

Another limitation with regards to geometrical structure are sharp corners, e.g. a rectangular one, which requires approximations such as the assumption of strong field confinement in the waveguide [8, 9]. In cylindrical coordinates (*r; φ)* the

> 0*n*2 ð Þ*r* � �Ψð Þ¼ r, φ 0 (5)

where *n r*ð Þ denotes the refractive index and Ψð Þ¼ *r*, *φ ψ*ð Þ*r* Φð Þ *φ* . *ψ*ð Þ*r* represents

At the bending area, it shows a higher tendency to produce radiation loss compared to straight waveguide due to the variance of phase front velocity correspond to the speed of light. These modes are strongly guided in the area with the highest refractive index. Thus, it will shift towards the outer surface of the bent waveguide. At the transition from straight to bent waveguide mode adaptation, losses will occur. Unfortunately, these losses are exponentially increased as we reduced the radii of curvature, which led to major drawbacks in developing nano-sized optical

Among quantum fictitious forces, the quantum anticentrifugal force which appears in the cylindrical symmetry system belongs to the most fascinating. This quantum phenomenon was first reported by [10] where he predicted that a free particle of vanishing angular momentum, *m* ¼ 0 experiences an attractive potential

towards the rotating centre rather than repulsive. The 2-dimension timeindependent Schrödinger equation of the free particle of mass, M and energy,

( )

ℏ2 ½ � *E* � *Vm*ð Þ*r*

ℏ2 2*M*

where *um*ð Þ*r* and *Vm*ð Þ*r* is a radial wave function and effective potential, respectively. This potential describes the existence of repulsive or attractive force on the free particle which is proportional to the value of *m*2. Under the non-vanishing angular momentum (*m* 6¼ 0Þ condition, it tends to repulse far from the rotating center thus, create the centrifugal force. On contrary, at zero angular momentum, its binding potential towards the rotating center become stronger and contribute to

The force corresponding to VQ cause a ring-shaped wave packet to spread in an asymmetric form where it spreads faster towards the origin. Consequently, the particle is localized on a band-shaped domain around the origin. On contrary,

*m*2 � 1*=*4

ℏ2 2*M*

1

*um*ð Þ¼ *r* 0 (6)

*r*2 (7)

4*r*2 (8)

2*M*

*Vm*ð Þ¼ *r*

the arising of quantum anti-centrifugal potential, *VQ*which is given by

*VQ* � *V*0ð Þ¼� *r*

solve the eigenmode problem analytically.

*DOI: http://dx.doi.org/10.5772/intechopen.94311*

*Quantum Signal over Optical Fiber*

Helmholtz equation takes the form of

**3. Quantum anti-centrifugal force**

E along the polar coordinates r and *φ* is given by

*d*2 *dr*2 þ

the bending mode profile.

waveguiding devices.

**21**

1 *r ∂ ∂r* þ *∂*2 *∂r* 2 þ 1 *r*2

$$
\nabla \times H - \varepsilon \frac{\partial E}{\partial t} = \mathbf{0} \tag{2}
$$

The permittivity tensor, *ε*, and permeability tensor, *μ* is uniform. Both of Eqs. (1) and (2) are satisfied by the following plane wave solution:

$$\Psi' = A e^{-i(\text{at} - k.r)} \tag{3}$$

where A and *ω* denotes an amplitude and angular frequency with the wavevector, j j *k* ¼ *ω* ffiffiffiffiffi *με* p .

However, for the anisotropic media such as Lithium niobate, calcite, and quartz, both *ε* and *μ* are directional dependent [3]. The permittivity tensor is derived by the following matrix expression:

$$
\boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}\_{\theta} \begin{bmatrix} n\_x^2 & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & n\_y^2 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & n\_x^2 \end{bmatrix} = \begin{bmatrix} \varepsilon\_x & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \varepsilon\_y & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \varepsilon\_x \end{bmatrix} \tag{4}$$

where *nx*, *ny* and *nz* are the principal indices of refraction whereas ð Þ *x*, *y*, *z* denotes the principal axis of the crystal. Generally, the anisotropic media can be divided into uniaxial and biaxial crystal. In uniaxial medium, the refractive index corresponds to two equal principle elements i.e. *n*2 *o* ¼ *εx=εo* ¼ *εy=εo* is defined as an ordinary index, *no* whereas the other index for the remaining principle (*n*2 *eo* ¼ *εz=εo*Þ is known extraordinary index.

The optical mode across the waveguiding area is derived from the abovementioned Maxwell's equation and Helmholtz propagation equation with a specific boundary condition. From this, it can show the deformation of spatial modes and optical loss in optical fiber [4–6]. The Helmholtz problem can be solved analytically when it is separable concerning its variables. However, this separability *Quantum Signal over Optical Fiber DOI: http://dx.doi.org/10.5772/intechopen.94311*

Apart from that, it is found that the bending of optical waveguide has led to the

signal distortion which increase the propagation losses. For the pass few years, researcher put an extra effort to observe this phenomena which is known an quantum anti-centrifugal force from a quantum physics point of view.

Therefore, the aim of this chapter lies on the discussion of poorly reviewed quantum anti-centrifugal force. Furthermore, we perceive the correlation of modern interpretation of Schrodinger equation with the classical Helmholtz equation correspond to this quantum fictitious force. The observation of this effect is extended into the real application of Lithium Niobate on Insulator rib waveguide as it found to be an extensive platfrom towards the development of quantum

The optical waveguide structure consists of high refractive index core, surrounded by slightly lower refractive index. It operates based on total internal reflection phenomena which allow the electromagnetic wave guiding towards core. The electromagnetic plane waves propagation in homogeneous and isotropic media

∇ � *E* þ *μ*

∇ � *H* � *ε*

where A and *ω* denotes an amplitude and angular frequency with the

*x* 0 0 0 *n*2

ordinary index, *no* whereas the other index for the remaining principle (*n*2

The optical mode across the waveguiding area is derived from the

*y* 0 0 0 *n*2

*z*

where *nx*, *ny* and *nz* are the principal indices of refraction whereas ð Þ *x*, *y*, *z* denotes the principal axis of the crystal. Generally, the anisotropic media can be divided into uniaxial and biaxial crystal. In uniaxial medium, the refractive index

abovementioned Maxwell's equation and Helmholtz propagation equation with a specific boundary condition. From this, it can show the deformation of spatial modes and optical loss in optical fiber [4–6]. The Helmholtz problem can be solved analytically when it is separable concerning its variables. However, this separability

However, for the anisotropic media such as Lithium niobate, calcite, and quartz, both *ε* and *μ* are directional dependent [3]. The permittivity tensor is derived by the

2 6 4

The permittivity tensor, *ε*, and permeability tensor, *μ* is uniform. Both of

*∂H*

*∂E*

*∂t* ¼ 0 (1)

*∂t* ¼ 0 (2)

*Ψ* ¼ *Ae*�*i*ð Þ ωt�*k:r* (3)

*εx* 0 0 0 *εy* 0 0 0 *εz* 3 7

5 (4)

*eo* ¼ *εz=εo*Þ

*o* ¼ *εx=εo* ¼ *εy=εo* is defined as an

such as glass and diamond are governed by Maxwell's equation:

Eqs. (1) and (2) are satisfied by the following plane wave solution:

information technology.

wavevector, j j *k* ¼ *ω* ffiffiffiffiffi

following matrix expression:

is known extraordinary index.

**20**

*με* p .

*ε* ¼ *εo*

corresponds to two equal principle elements i.e. *n*2

*n*2

2 6 4

**2. Optical field propagation**

*Application of Optical Fiber in Engineering*

does not hold for all coordinate systems [7]. Therefore, the need to choose for a suitable coordinate system and boundary conditions are crucially needed prior to solve the eigenmode problem analytically.

Another limitation with regards to geometrical structure are sharp corners, e.g. a rectangular one, which requires approximations such as the assumption of strong field confinement in the waveguide [8, 9]. In cylindrical coordinates (*r; φ)* the Helmholtz equation takes the form of

$$\left(\frac{1}{r}\frac{\partial}{\partial r} + \frac{\partial^2}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2}{\partial \rho^2} + k\_0^2 n^2(r)\right)\Psi(\mathbf{r}, \mathbf{q}) = \mathbf{0} \tag{5}$$

where *n r*ð Þ denotes the refractive index and Ψð Þ¼ *r*, *φ ψ*ð Þ*r* Φð Þ *φ* . *ψ*ð Þ*r* represents the bending mode profile.

At the bending area, it shows a higher tendency to produce radiation loss compared to straight waveguide due to the variance of phase front velocity correspond to the speed of light. These modes are strongly guided in the area with the highest refractive index. Thus, it will shift towards the outer surface of the bent waveguide. At the transition from straight to bent waveguide mode adaptation, losses will occur. Unfortunately, these losses are exponentially increased as we reduced the radii of curvature, which led to major drawbacks in developing nano-sized optical waveguiding devices.

### **3. Quantum anti-centrifugal force**

Among quantum fictitious forces, the quantum anticentrifugal force which appears in the cylindrical symmetry system belongs to the most fascinating. This quantum phenomenon was first reported by [10] where he predicted that a free particle of vanishing angular momentum, *m* ¼ 0 experiences an attractive potential towards the rotating centre rather than repulsive. The 2-dimension timeindependent Schrödinger equation of the free particle of mass, M and energy, E along the polar coordinates r and *φ* is given by

$$\left\{\frac{d^2}{dr^2} + \frac{2M}{\hbar^2} [E - V\_m(r)]\right\} u\_m(r) = 0\tag{6}$$

$$V\_m(r) = \frac{\hbar^2}{2M} \frac{m^2 - 1/4}{r^2} \tag{7}$$

where *um*ð Þ*r* and *Vm*ð Þ*r* is a radial wave function and effective potential, respectively. This potential describes the existence of repulsive or attractive force on the free particle which is proportional to the value of *m*2. Under the non-vanishing angular momentum (*m* 6¼ 0Þ condition, it tends to repulse far from the rotating center thus, create the centrifugal force. On contrary, at zero angular momentum, its binding potential towards the rotating center become stronger and contribute to the arising of quantum anti-centrifugal potential, *VQ*which is given by

$$V\_Q \equiv V\_0(r) = -\frac{\hbar^2}{2M} \frac{1}{4r^2} \tag{8}$$

The force corresponding to VQ cause a ring-shaped wave packet to spread in an asymmetric form where it spreads faster towards the origin. Consequently, the particle is localized on a band-shaped domain around the origin. On contrary,

Cirone claimed that this potential force has not existed in 3-dimensional space. However, in 2010, Dandoloff proves that this counterintuitive potential existed in 3-dimensional space on wormhole geometry [11].

In 2011, he reported his finding on quantum anti-centrifugal potential dependences over the radius of the bent rectangular waveguide as depicted in **Figure 1** [12, 13]. Based on the following line element expression, the Schrödinger equation of this Cartesian coordinate can be derived as Eq. (10).

$$dr^2 = d\xi^2 + dz^2 + \left(\mathbf{1} - \kappa\xi\right)^2 ds^2 \tag{9}$$

In 2015, Victor published another finding on quantum anti-centrifugal potential by introducing half-geodesic coordinates where the curvature with constant radius corresponds to the concentric circles at *r* = 0 [7]. Its 2D Laplacian function is

> þ *∂ ∂ϕ*

1 *h ∂h ∂ϕ ψ* (14)

*c*2 and *K* ¼ �3 *ω*2

*r*2 � *m*2 � 1 *ω*2

(15)

*c*2 (16)

(17)

*c*2 , Eq. (14) can be

*∂*2 *rh* and

*∂rh*

*Δψ* ¼ 1 *h* *∂h ∂r*

*V* ¼ ℏ2 2*M*

*c* ≪ 1, *kg* ¼ 1

*V* ¼ � ℏ2 2*M*

> *V* ¼ ℏ2 2*M*

centrifugal states additional legitimacy [14, 15].

**quantum anti-centrifugal force**

*h ∂h ∂r*

In this work, an intrinsic Gaussian curvature of the surface, *K* ¼ �*h*�1

the geodesic curvature of the geodesically parallel coordinates lines *kg* ¼ �*h*�1

has been introduced. Thus, the effective potential, V may be written as follows:

*m*2 *h*2 � 1 4 *k*2 *g* � 1 2*K*

*r*2 þ 2 *ω*2

*m*2 � 1 4

Eq. (15) describes the anti-centrifugal potential in a 2D plane at any surface which can be covered globally with a geodesically parallel coordinate system. At *K* > 0 (i.e elliptical), the curvature weakens the anti-centrifugal potential whereas at *K* < 0 (i.e catenoid), its potential enhanced. Recently, Dandoloff has included the quantum geometric potential (QGP) which commonly occurs at the surface of the structure, using a very high sandwich potential around the surface. The total effec-

> *m*2 *h*2 � 1 4 *k*2 *g* � 1 2*K*

**4. Schrödinger vs. Helmholtz equation towards the measurement of**

As of today, the investigation of the quantum anti-centrifugal potential was investigated purely theoretical correspond to the Schrödinger equation. However, the advancement of the nano-fabrication and quantum photonics technology since last decides has brought the impossible into reality. It has become a thriving field of research in promoting both fundamental studies of quantum phenomena and a wider range of disruptive quantum technologies. The well-developed quantum experimental setting such as the phase retrieval algorithm and single-photon detection techniques has brought the experimental on quantum anti-centrifugal potential into reality. Recently, a lot of studies on spatial mode deformation and losses in optical fiber in various geometrical structure has been carried out to observe the

From this research, Dandoloff addressed his new finding on the total quantum effective potential which appears to be the nonlinear sigma model plus positive terms instead of the negative signage. Even though the exact number of states cannot be defined by using this Topologically Stable States, but it provides the

expressed as:

expressed as:

Within the limit of *ωr*

*Quantum Signal over Optical Fiber*

*DOI: http://dx.doi.org/10.5772/intechopen.94311*

tive potential is given by:

**23**

$$
\Delta\Psi = \frac{\partial^2 \Psi}{\partial \mathbf{z}^2} + \frac{\delta^2 \Psi}{\delta \xi^2} - \frac{\kappa}{(1 - \kappa \xi)} \frac{\delta \Psi}{\delta \xi} + \frac{1}{(1 - \kappa \xi)^2} \frac{\delta^2 \Psi}{\delta \xi^2} \tag{10}
$$

where *ξ*∈ � *a* 2, *a* 2 � �, *κ* ¼ 1 *R*, *C* ¼ *C a*ð Þ , *R* and Ψ ¼ 1ffiffi *C* p Φ 1�*κξ*.

The general time-independent Schrödinger equation of *i*ℏ2 2*M* ΔΨ ¼ EΨ is then can be expressed in ð Þ *ξ*, *s*, *z* coordinate system as:

$$E\frac{\Phi}{(1-\kappa\xi)^{\frac{1}{2}}} = \frac{i\hbar^2}{2M} \left[ \frac{1}{(1-\kappa\xi)^{\frac{1}{2}}} \frac{\delta^2\Phi}{\delta\xi^2} + \frac{k^2}{4} \frac{\Phi}{(1-\kappa\xi)^{\frac{2}{2}}} + \frac{\partial^2\Phi}{\partial\mathbf{z}^2} + \frac{1}{(1-\kappa\xi)^2} \frac{\delta^2\Phi}{\delta\mathbf{s}^2} \right] \tag{11}$$

Compared to Eq. (8), Dandoloff proposed the Bohm potential, *Q* corresponding to the bent waveguide which is given by

$$Q = -\frac{\hbar^2}{2M} \frac{\Delta R}{R} \tag{12}$$

where R is the radius of the waveguide. This force was assumed to affect the quantum motion on a bent waveguide. However, he found that this effect has a very weak contribution to the overall potential. Thus, the quantum anti-centrifugal potential is then derived as:

$$-\frac{\hbar^2}{2M} \frac{dV\_{\text{eff}}}{d\xi|\_{\xi=0}} = \frac{\hbar^2}{2M} \frac{\kappa^3}{2} = \frac{\hbar^2}{2M} \frac{1}{2R^3} \tag{13}$$

It is found that at zero angular momentum (m = 0), the anti-centrifugal reciprocally dependence on R3 . Dandaloff also concluded the similarity of the stationary Schrödinger equation and Helmholtz equation for the TE modes in a waveguide structure. Subsequently, the interference pattern in the quantum medium should have the same pattern as an electromagnetic wave in a bent waveguide.

**Figure 1.** *Semi-circular shaped bent waveguide on Cartesian coordinates, (*x,y,z*).*

*Quantum Signal over Optical Fiber DOI: http://dx.doi.org/10.5772/intechopen.94311*

Cirone claimed that this potential force has not existed in 3-dimensional space. However, in 2010, Dandoloff proves that this counterintuitive potential existed in

In 2011, he reported his finding on quantum anti-centrifugal potential dependences over the radius of the bent rectangular waveguide as depicted in **Figure 1** [12, 13]. Based on the following line element expression, the Schrödinger equation

*dr*2 ¼ *dξ*2 þ *dz*2 þ ð Þ 1 � *κξ* 2

*R*, *C* ¼ *C a*ð Þ , *R* and Ψ ¼ 1ffiffi

*k*2 4

*Q* ¼ � ℏ2 2*M*

where R is the radius of the waveguide. This force was assumed to affect the quantum motion on a bent waveguide. However, he found that this effect has a very weak contribution to the overall potential. Thus, the quantum anti-centrifugal

> ¼ ℏ2 2*M κ*3 2 ¼ ℏ2 2*M*

It is found that at zero angular momentum (m = 0), the anti-centrifugal recip-

Schrödinger equation and Helmholtz equation for the TE modes in a waveguide structure. Subsequently, the interference pattern in the quantum medium should

Φ ð Þ 1 � *κξ* 5 2 þ *∂*2 Φ *∂*z2 þ

Compared to Eq. (8), Dandoloff proposed the Bohm potential, *Q* corresponding

" #

Δ*R*

ð Þ 1 � *κξ*

*δ*Ψ *δξ* þ

> *C* p Φ 1�*κξ*.

1 ð Þ 1 � *κξ* 2 *δ*2 Ψ

1 ð Þ 1 � *κξ* 2

*R* (12)

2*R*3 (13)

1

. Dandaloff also concluded the similarity of the stationary

*ds*2 (9)

*δξ*2 (10)

2*M* ΔΨ ¼ EΨ is then can

(11)

*δ*2 Φ *δs*2

3-dimensional space on wormhole geometry [11].

*Application of Optical Fiber in Engineering*

ΔΨ ¼ *∂*2

be expressed in ð Þ *ξ*, *s*, *z* coordinate system as:

to the bent waveguide which is given by

potential is then derived as:

rocally dependence on R3

**Figure 1.**

**22**

2, *a* 2 � �, *κ* ¼ 1

where *ξ*∈ � *a*

*E* Φ ð Þ 1 � *κξ* 1 2 ¼ *i*ℏ2 2*M*

of this Cartesian coordinate can be derived as Eq. (10).

Ψ *∂*z2 þ

1 ð Þ 1 � *κξ* 1 2 *δ*2 Φ *δξ*2 þ

> � ℏ2 2*M*

*Semi-circular shaped bent waveguide on Cartesian coordinates, (*x,y,z*).*

*dVeff dξ*j *ξ*¼0

have the same pattern as an electromagnetic wave in a bent waveguide.

*δ*2 Ψ *δξ*2 � *κ*

The general time-independent Schrödinger equation of *i*ℏ2

In 2015, Victor published another finding on quantum anti-centrifugal potential by introducing half-geodesic coordinates where the curvature with constant radius corresponds to the concentric circles at *r* = 0 [7]. Its 2D Laplacian function is expressed as:

$$
\Delta\psi = \frac{1}{h} \left[ \frac{\partial h}{\partial r} \left( h \frac{\partial h}{\partial r} \right) + \frac{\partial}{\partial \phi} \left( \frac{1}{h} \frac{\partial h}{\partial \phi} \right) \right] \psi \tag{14}
$$

In this work, an intrinsic Gaussian curvature of the surface, *K* ¼ �*h*�1 *∂*2 *rh* and the geodesic curvature of the geodesically parallel coordinates lines *kg* ¼ �*h*�1 *∂rh* has been introduced. Thus, the effective potential, V may be written as follows:

$$V = \frac{\hbar^2}{2M} \left( \frac{m^2}{h^2} - \frac{1}{4}k\_g^2 - \frac{1}{2K} \right) \tag{15}$$

Within the limit of *ωr c* ≪ 1, *kg* ¼ 1 *r*2 þ 2 *ω*2 *c*2 and *K* ¼ �3 *ω*2 *c*2 , Eq. (14) can be expressed as:

$$V = -\frac{\hbar^2}{2M} \left( \frac{m^2 - \frac{1}{4}}{r^2} - \left( m^2 - 1 \right) \frac{\alpha^2}{c^2} \right) \tag{16}$$

Eq. (15) describes the anti-centrifugal potential in a 2D plane at any surface which can be covered globally with a geodesically parallel coordinate system. At *K* > 0 (i.e elliptical), the curvature weakens the anti-centrifugal potential whereas at *K* < 0 (i.e catenoid), its potential enhanced. Recently, Dandoloff has included the quantum geometric potential (QGP) which commonly occurs at the surface of the structure, using a very high sandwich potential around the surface. The total effective potential is given by:

$$V = \frac{\hbar^2}{2M} \left( \frac{m^2}{h^2} - \frac{1}{4}k\_g^2 - \frac{1}{2K} \right) \tag{17}$$

From this research, Dandoloff addressed his new finding on the total quantum effective potential which appears to be the nonlinear sigma model plus positive terms instead of the negative signage. Even though the exact number of states cannot be defined by using this Topologically Stable States, but it provides the centrifugal states additional legitimacy [14, 15].

## **4. Schrödinger vs. Helmholtz equation towards the measurement of quantum anti-centrifugal force**

As of today, the investigation of the quantum anti-centrifugal potential was investigated purely theoretical correspond to the Schrödinger equation. However, the advancement of the nano-fabrication and quantum photonics technology since last decides has brought the impossible into reality. It has become a thriving field of research in promoting both fundamental studies of quantum phenomena and a wider range of disruptive quantum technologies. The well-developed quantum experimental setting such as the phase retrieval algorithm and single-photon detection techniques has brought the experimental on quantum anti-centrifugal potential into reality. Recently, a lot of studies on spatial mode deformation and losses in optical fiber in various geometrical structure has been carried out to observe the

similar quantum effect by using the Helmholtz equation with a suitable boundary condition.

of *neff* contribute to the eigenmode (fundamental) solution of Maxwell's equation. However, if the condition is not met (*neff* < *ns*), the confined mode is degenerated and produced the radiating mode which contributes to the increment of propagation losses. Fortunately, the mode profile of the LNOI rib waveguide shows a prominent similarity with the toroidal waveguide structure despite the radius variation as depicted in **Figure 3(b)**. This output shows a strong consistency with the analytical solution in [17]. However, the slight asymmetric of the mode profile is led by the natural consequences of the symmetry breaking due to the presence of the

The aim of this chapter is to extend the above analysis by considering the variation of a radius on the rib waveguide prior to observe the deformation pattern of the mode profile within the bending area. Thus, the bending radius is set from 5 to 90 μm at a fixed width of the core value of 1 μm. From **Table 1**, it shows that the average radial position reduced as we c, which is inline with the finding in [17]. Consequently, the number of higher order mode increased as we increase the bending radius. However, at higher bending radius, the mode profile shows the symmetrical pattern compared to lower bending radius which tend to produce the

Based on **Table 2**, the effective refractive index value decreased which reflect the decrement of propagation constant value. Simultaneously, the bending loss is increase drastically as the bending radius increase. The value is contributed by the summation of radiation, absorption, and scattering losses. However, it is mainly contributed by the scattering losses which occur due to the spatial fluctuations of the refractive index or the sidewalls roughness of the waveguide and radiation loss

> **Average radial position,** *rav***> (μm) R=5 μm R = 10 μm R = 30 μm R = 50 μm R = 70 μm R = 90 μm** *l=1 l=2 l=1 l=2 l=1 l=2 l=1 l=2 l=1 l=2 l=1 l=2*

**Effective refractive index,** *neff* **R=5 μm R = 10 μm R = 30 μm R = 50 μm R = 70 μm R = 90 μm** *l=1 l=2 l=1 l=2 l=1 l=2 l=1 l=2 l=1 l=2 l=1 l=2*

*n=1* 2.452 2.1574 2.331 2.125 2.223 2.022 2.207 2.012 2.201 2.000 2.197 1.997 *n=2* 2.271 — 2.2087 2.0014 2.123 1.8542 2.096 1.842 2.090 1.837 2.087 1.834 *n=3* 2.101 — 2.0603 2.0603 1.962 — 1.950 — 1.945 — 1.942 *n=4* — — 1.8357 1.748 — 1.737 — 1.732 — 1.729 —

*n=1* 1.199 1.3237 1.133 1.126 1.051 1.047 1.031 1.022 1.023 1.019 1.018 1.016 *n=2* 1.014 — 0.977 0.982 0.987 1.00 0.988 1.102 0.997 1.003 0.995 1.002 *n=3* 1.016 — 0.972 — 0.981 — 0.997 — 0.991 — 0.998 *n=4* — — 0.900 — 0.995 — 0.996 — 0.980 — 0.900 —

asymmetric profile towards the outer surface of the waveguide.

*The average radial position of LNOI rib waveguide for various bent radius.*

*The effective refractive index of LNOI rib waveguide for various bent radius.*

due to the imperfect mode guiding across the waveguide (**Table 2**) [18].

oxide layer.

*Quantum Signal over Optical Fiber*

*DOI: http://dx.doi.org/10.5772/intechopen.94311*

**Mode number**

**Table 1.**

**Mode number**

**Table 2.**

**25**

Recently, the deformation of eigenmode in the semi-circular toroidal waveguide is reported in [17] as depicted in **Figure 2(a)** by using the finite element method. The operating wavelength and refractive index of both waveguides, *nw* and surrounding, *ns* was set at 800 nm and 2.3 and 1.0, respectively. The mode average radial position was introduced in this simulation work in order to analyze the direction of mode. The bending radius was set at 1 μm. On top of that, the analysis was extended to a realistic structure and promising material in quantum photonic technology, Lithium Niobate on insulator (LNOI) rib waveguide [16]. Lithium Niobate (LN) is has a strong nonlinear electro-optical property which transparent over a broad spectrum spanning from visible to mid-infrared region (350 nm to 5200 nm).

**Figure 2(b)** depicted the rib structure which consists of 500 um thick LN substrate, 2 μm thick SiO2 which acts as a buffer layer, and a 600 nm thin film LN. The thin film LN is etched to form the higher confinement core with height, D, and width, W of 500 nm and 1000 nm, respectively. This simulation was performed in an axial-symmetric profile at a second harmonic wavelength of 775 nm.

Based on **Figure 3(a)**, the spatial mode distorted and shifted towards the outer wall of the bent waveguide. As the mode propagates across the waveguide, it is strongly guided towards the area with a higher refractive index that leads to the mode shifting towards the outer rim of the bent waveguide. The maximum higherorder propagating mode of (2,2) was obtained with the higher intensity are found to be closer towards the inner wall of the waveguide. The propagating mode must fulfill this condition: *ns* <*neff* <*nw* where *neff* denotes the effective refractive index. This value describes the propagation characteristic of the mode which proportional to propagation constant, β but reciprocal to wavenumber, *κo*. The maximum value

#### **Figure 2.**

*(a) Toroidal waveguide with a width of core, a* ¼ *r*2 � *r*1 ¼ 1 *μm and height, b* ¼ 0*:*5 *μm where r*1 ¼ 0*:*5 *μm and r*2 ¼ 1*:*5 *μm. (b) LNOI rib waveguide with the thickness of the slab, h* ¼ *H* � *D [17].*

**Figure 3.** *Mode profile of bent waveguide in (a) toroidal and (b) LNOI rib structure [17].*

### *Quantum Signal over Optical Fiber DOI: http://dx.doi.org/10.5772/intechopen.94311*

similar quantum effect by using the Helmholtz equation with a suitable boundary

**Figure 2(b)** depicted the rib structure which consists of 500 um thick LN substrate, 2 μm thick SiO2 which acts as a buffer layer, and a 600 nm thin film LN. The thin film LN is etched to form the higher confinement core with height, D, and width, W of 500 nm and 1000 nm, respectively. This simulation was performed in

Based on **Figure 3(a)**, the spatial mode distorted and shifted towards the outer wall of the bent waveguide. As the mode propagates across the waveguide, it is strongly guided towards the area with a higher refractive index that leads to the mode shifting towards the outer rim of the bent waveguide. The maximum higherorder propagating mode of (2,2) was obtained with the higher intensity are found to be closer towards the inner wall of the waveguide. The propagating mode must fulfill this condition: *ns* <*neff* <*nw* where *neff* denotes the effective refractive index. This value describes the propagation characteristic of the mode which proportional to propagation constant, β but reciprocal to wavenumber, *κo*. The maximum value

*(a) Toroidal waveguide with a width of core, a* ¼ *r*2 � *r*1 ¼ 1 *μm and height, b* ¼ 0*:*5 *μm where r*1 ¼ 0*:*5 *μm*

*and r*2 ¼ 1*:*5 *μm. (b) LNOI rib waveguide with the thickness of the slab, h* ¼ *H* � *D [17].*

*Mode profile of bent waveguide in (a) toroidal and (b) LNOI rib structure [17].*

an axial-symmetric profile at a second harmonic wavelength of 775 nm.

from visible to mid-infrared region (350 nm to 5200 nm).

Recently, the deformation of eigenmode in the semi-circular toroidal waveguide is reported in [17] as depicted in **Figure 2(a)** by using the finite element method. The operating wavelength and refractive index of both waveguides, *nw* and surrounding, *ns* was set at 800 nm and 2.3 and 1.0, respectively. The mode average radial position was introduced in this simulation work in order to analyze the direction of mode. The bending radius was set at 1 μm. On top of that, the analysis was extended to a realistic structure and promising material in quantum photonic technology, Lithium Niobate on insulator (LNOI) rib waveguide [16]. Lithium Niobate (LN) is has a strong nonlinear electro-optical property which transparent over a broad spectrum spanning

condition.

*Application of Optical Fiber in Engineering*

**Figure 2.**

**Figure 3.**

**24**

of *neff* contribute to the eigenmode (fundamental) solution of Maxwell's equation. However, if the condition is not met (*neff* < *ns*), the confined mode is degenerated and produced the radiating mode which contributes to the increment of propagation losses. Fortunately, the mode profile of the LNOI rib waveguide shows a prominent similarity with the toroidal waveguide structure despite the radius variation as depicted in **Figure 3(b)**. This output shows a strong consistency with the analytical solution in [17]. However, the slight asymmetric of the mode profile is led by the natural consequences of the symmetry breaking due to the presence of the oxide layer.

The aim of this chapter is to extend the above analysis by considering the variation of a radius on the rib waveguide prior to observe the deformation pattern of the mode profile within the bending area. Thus, the bending radius is set from 5 to 90 μm at a fixed width of the core value of 1 μm. From **Table 1**, it shows that the average radial position reduced as we c, which is inline with the finding in [17]. Consequently, the number of higher order mode increased as we increase the bending radius. However, at higher bending radius, the mode profile shows the symmetrical pattern compared to lower bending radius which tend to produce the asymmetric profile towards the outer surface of the waveguide.

Based on **Table 2**, the effective refractive index value decreased which reflect the decrement of propagation constant value. Simultaneously, the bending loss is increase drastically as the bending radius increase. The value is contributed by the summation of radiation, absorption, and scattering losses. However, it is mainly contributed by the scattering losses which occur due to the spatial fluctuations of the refractive index or the sidewalls roughness of the waveguide and radiation loss due to the imperfect mode guiding across the waveguide (**Table 2**) [18].

**Table 1.**

*The average radial position of LNOI rib waveguide for various bent radius.*

**Table 2.**

*The effective refractive index of LNOI rib waveguide for various bent radius.*


**3. A Case of nuclear analytical application in animal sciences: Uranium in rock phosphates and rabbit muscles from animals receiving uranium determined by neutron activation analysis**

Sedimentary ore deposits have provided about 80%–90% of world production of phosphate [6]. Oppositely, in Brazil igneous deposits represent 80% of the national reserves of phosphate rocks. The major Brazilian phosphate rock reserves are concentrated in the states of Minas

Nuclear Analytical Techniques in Animal Sciences: New Approaches and Outcomes

http://dx.doi.org/10.5772/55071

183

In 2009, the Brazilian phosphate-related products market was shared as phosphate fertilizers (89.12%), animal feed – mainly dicalcium phosphate (6.91%), soil amendment (1.01%), and the

**Ingredient (tons.year-1)** *2005 2011 2012***∇**

Dicalcium Phosphate 216.400 404.761 560.274

Limestone 634.000 1.247.016 1.276.388

Bone Meal 225.400 n.i. n.i.

**Products Quantity (tons) FOB x U\$1,000.00**

United States n.a. 33.31 30.59 Algeria 20.90 n.a. n.a. Morocco 63.80 24.54 27.74 Israel 10.30 8.97 9.08 Tunisia 4.71 7.95 7.68 Russia n.a. 6.74 6.19 Spain 0.13 n.a. n.a. Others 0.16 18.49 18.71

**Table 3.** Brazilian imports of phosphates by countries, raw and industrialized products. Total pondered by the price

**Industrialized Material (%)**

**Total (%)**

Raw material 915.449 84.040 Industrialized 2.861.719 945.170 Total 3.777.168 1.029.210

Gerais, Goiás and São Paulo [17].

remaining was not informed (2.96%) [17].

n.i., not informed., ∇ forecasted in 2012, May for the entire year

**Table 1.** Phosphorus and calcium sources used in the Brazilian Feed Industry [8]

**Table 2.** Brazilian imports of phosphates, raw and industrialized products [17]

**Raw Material (%)**

**Country**

FOB, DNPM [17]

Nuclear applications in Animal Sciences are not a novelty. It could be considered as the two first studies from 1949 using rabbits to assess the biological consequences of the uranium ingestion: Decreased body weight was reported for rabbits exposed to 11 mg U/ m3 as uranium tetrachloride dust for 35–40 days [9]. Rabbits lost 22% of their body weight during a 30 days exposure to 0.9 mg U/m3 , dogs and cats lost approximately 25% of their body weight during a similar exposure to 9.5 mg U/m3 . Similar effects were observed with uranium tetrafluoride [10].

**Introduction**: Phosphorus (P) deficiency in crops is an important constraining factor n plant and animal yields, especially in hot humid tropics where soils are predominantly acidic and often extremely P deficient with high P fixation capacities [11,12].

Phosphorus combines with oxygen forming oxides called phosphates. Phosphates are defined as compounds, which contain phosphorus-oxygen (P-O) linkages. The P-O bond has a length of 1.62Å with bond angles of 130° at the oxygen atoms and 102° at the phosphorus atoms at the pentoxide P2O5, the only oxide of phosphorus that is produced commercially [13].

Phosphate rock denotes the product obtained from the mining and subsequent metallurgical processing of phosphorus bearing ores. By flotation of phosphate rock it is formed apatite concentrates. These phosphate products are the major phosphorus sources in soil nutrition, also plenty used in animal formulations by industries [14].

*they cremated some one. Now they recover over ninety-eight per cent of it.  Which makes*

*the best part of four hundred tons of phosphorus every year from England alone." Henry spoke with a happy pride, rejoicing whole-*

*heartedly in the achievement, as though it had been his own. "Fine to think we can go on being socially useful even after we're dead.*

*Making plants grow.'…} Brave New World, Aldous Leonard Huxley [15].*

This text above was extracted from the 1932 fictional worldwide bestseller *Brave New World* [15]. It could sound so fantasist back when it was firstly published. Maybe it will sound reality in a future not far ahead. Currently, a reduced and shortening number of phosphate sites are mined around the world. Based on current phosphate extraction rates and economic trends in the 1990s, more than half of main phosphate producer countries will have exceeded the life of their reserves up to 2024 [16].

Phosphate ranks second (coal and hydrocarbons excluded) in terms of gross tonnage and volume of international trade [16].

*…'On their way up the chimney the gases go through four separate treatments. P2O5 used to go right out of circulation every time*

Sedimentary ore deposits have provided about 80%–90% of world production of phosphate [6]. Oppositely, in Brazil igneous deposits represent 80% of the national reserves of phosphate rocks. The major Brazilian phosphate rock reserves are concentrated in the states of Minas Gerais, Goiás and São Paulo [17].

In 2009, the Brazilian phosphate-related products market was shared as phosphate fertilizers (89.12%), animal feed – mainly dicalcium phosphate (6.91%), soil amendment (1.01%), and the remaining was not informed (2.96%) [17].

n.i., not informed., ∇ forecasted in 2012, May for the entire year

**3. A Case of nuclear analytical application in animal sciences: Uranium in**

Nuclear applications in Animal Sciences are not a novelty. It could be considered as the two first studies from 1949 using rabbits to assess the biological consequences of the uranium ingestion: Decreased body weight was reported for rabbits exposed to 11 mg U/ m3 as uranium tetrachloride dust for 35–40 days [9]. Rabbits lost 22% of their body weight

**Introduction**: Phosphorus (P) deficiency in crops is an important constraining factor n plant and animal yields, especially in hot humid tropics where soils are predominantly acidic and

Phosphorus combines with oxygen forming oxides called phosphates. Phosphates are defined as compounds, which contain phosphorus-oxygen (P-O) linkages. The P-O bond has a length of 1.62Å with bond angles of 130° at the oxygen atoms and 102° at the phosphorus atoms at

Phosphate rock denotes the product obtained from the mining and subsequent metallurgical processing of phosphorus bearing ores. By flotation of phosphate rock it is formed apatite concentrates. These phosphate products are the major phosphorus sources in soil nutrition,

*…'On their way up the chimney the gases go through four separate treatments. P2O5 used to go right out of circulation every time*

*they cremated some one. Now they recover over ninety-eight per cent of it.  Which makes*

*the best part of four hundred tons of phosphorus every year from England alone." Henry spoke with a happy pride, rejoicing whole-*

*heartedly in the achievement, as though it had been his own. "Fine to think we can go on being socially useful even after we're dead.*

This text above was extracted from the 1932 fictional worldwide bestseller *Brave New World* [15]. It could sound so fantasist back when it was firstly published. Maybe it will sound reality in a future not far ahead. Currently, a reduced and shortening number of phosphate sites are mined around the world. Based on current phosphate extraction rates and economic trends in the 1990s, more than half of main phosphate producer countries will have exceeded the life of

Phosphate ranks second (coal and hydrocarbons excluded) in terms of gross tonnage and

the pentoxide P2O5, the only oxide of phosphorus that is produced commercially [13].

, dogs and cats lost approximately 25% of their

. Similar effects were observed with

**rock phosphates and rabbit muscles from animals receiving uranium**

182 Imaging and Radioanalytical Techniques in Interdisciplinary Research - Fundamentals and Cutting Edge Applications

**determined by neutron activation analysis**

body weight during a similar exposure to 9.5 mg U/m3

often extremely P deficient with high P fixation capacities [11,12].

also plenty used in animal formulations by industries [14].

*Making plants grow.'…} Brave New World, Aldous Leonard Huxley [15].*

their reserves up to 2024 [16].

volume of international trade [16].

during a 30 days exposure to 0.9 mg U/m3

uranium tetrafluoride [10].

**Table 1.** Phosphorus and calcium sources used in the Brazilian Feed Industry [8]

**Table 2.** Brazilian imports of phosphates, raw and industrialized products [17]

**Table 3.** Brazilian imports of phosphates by countries, raw and industrialized products. Total pondered by the price FOB, DNPM [17]

Data in table 1 demonstrate the economic relevance of the products under consideration in this study for the Brazilian feed industry. Foreign products are also of concern, as long Brazilian crops and animal yields rely on a great amount of imported phosphate prod‐ ucts (tables 2 and 3).

Practically every rock phosphate contains uranium in its structure [16,24]. The amounts of this and others hazardous substances vary widely among phosphates sources and it may vary even in the same deposit. Thus, mining, milling, industrializing and using phosphate products in soil and animal nutrition are anthropogenic activities increasing the potential for human

German studies in the 1970´s pointed-out the evidence for a raising uranium presence in rivers and groundwater in regions with intensive use of phosphate fertilizers in agriculture (figure 1). Uranium derived from phosphate fertilizers is likely to be adsorbed on the uppermost soil

Uranium content can be determined by the nuclear method of the Neutron Activation Analysis. This is a precise, fast (short turn-around), sensitive and non-destructive method [2].

*Phosphate sample preparation*: phosphates were acquired in the local market of Minas Gerais. Aliquots of 100 grams were randomly taken for each product pack to be grounded to obtain a particle size of 200 Tyler mesh (75 μm) establishing similar conditions for all samples (99% of conformity of the particle size of each product). Aliquot of 1000 mg of each grounded product

*Animal breeding:* Two groups of twelve (6 females and 6 males) New Zealand white rabbits (30 days of life) were selected and separated in two groups housed individually receiving a

Two rabbit feeds were designed to allow the introduction of the P source and to offer sufficient nutrient intake to meet rabbit nutritive requirements. Each one of the feeds had the same 98 percent (dry basis) of fiber, energy, and amino acids. Both formulations were based in raw materials as %, dry basis: Alfalfa meal 34.63, Soy oil 1.00, Sugar cane 2.00, Salt 0.50, Lysine 0.25, Methionine 0.04, Limestone 1.0, Premix 0.40, Maize 6.05, Wheat straw 25.0, Soybean meal 12.13, Maize by products 15.00 and the remaining 2.0 from the selected P source – dicalcium

Feeds were processed in order to turn the mixed products into a compact mixture. After that, the meal was conditioned by mixing it on dry steam in a conditioner; this conditioned product was pressed by rolls to pass through the holes of the pelleting die which shapes the meal into the final pellet shape of 3.00 mm to permit a good balance between pellet quality and good

Animals were housed in stainless steel cages with a fenestrated floor to allow feces to drop through into a pan. Absorbent material was placed in the pan to collect urine and minimize

Good quality water was provided through a nipple-drinking system that provides water at all times. Food was provided by a J-hopper attached to the front of the cage. J-hopper

different phosphorus source (dicalcium phosphate and bovine bone meal).

phosphate and bone meal, both materials analyzed in first part of the experiment.

Nuclear Analytical Techniques in Animal Sciences: New Approaches and Outcomes

content in the river [25].

http://dx.doi.org/10.5772/55071

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layers and its content on the water is correlated to the HCO3

was weighted and sealed in small polystyrene capsules.

ammonia release due to the bacterial breakdown of urea.

exposure to uranium [24,25].

**4. Material and procedures**

intestinal motility.

Indeed, any phosphate mined worldwide may contain accessory-gangue minerals and impurities that can be hazardous to man and animal such: Cd, Hg, Pb and V [16].

The United States Agency for Toxic Substances and Disease Registry [19] publishes the CERCLA Priority List of Hazardous Substances that includes substances, which have been determined to be of the greatest public health concern [19]. Uranium is the 97th substance ranked in the list (table 4).

\*The ranking of hazardous substances on the CERCLA Priority List is based on three criteria (i,ii,iii). They form altogether the Total Score = Σ (i) Frequency of Occurrence + (ii) Toxicity + (iii) Potential for Human Exposure = Σ (i) up to 1.800 Points + (ii) up to 600 points + (iii) up to [300 concentration points + 300 exposure points] [20]

**Table 4.** Compilation of some hazardous substances (including uranium) in the CERCLA List [19, 20]

This study deals with the uranium, since to face the main constraints low inherent P in soil and plants, rock phosphates are likely to be more extensively disseminated in the agriculture and these phosphorus sources carry uranium from their structures to the environment and human food chain as well [21].

Uranium is the heaviest natural element in the nature; it is hazardous element in man and animal health, not just it presents radioactivity but also it presents metallotoxicity once it is a heavy metal. Furthermore, uranium presents many radionuclides with high radioactiv‐ ity and energy [22].

Health implications of human exposure to uranium are well documented: cancer, liver and kidney diseases and reproduction impairment [21,22].

Uranium in nature is more plentiful than silver (Ag) and about as abundant as arsenic (As). It is found in very small amounts in the form of minerals, especially in rocks, soil, water, air, plants and animal tissues that could be consumed as food containing varying amounts of uranium [23].

Practically every rock phosphate contains uranium in its structure [16,24]. The amounts of this and others hazardous substances vary widely among phosphates sources and it may vary even in the same deposit. Thus, mining, milling, industrializing and using phosphate products in soil and animal nutrition are anthropogenic activities increasing the potential for human exposure to uranium [24,25].

German studies in the 1970´s pointed-out the evidence for a raising uranium presence in rivers and groundwater in regions with intensive use of phosphate fertilizers in agriculture (figure 1). Uranium derived from phosphate fertilizers is likely to be adsorbed on the uppermost soil layers and its content on the water is correlated to the HCO3 content in the river [25].

Uranium content can be determined by the nuclear method of the Neutron Activation Analysis. This is a precise, fast (short turn-around), sensitive and non-destructive method [2].