In order to discuss the application and mode of the forced modulation function in a sensor, the optical fibre emphasis function was established by referring to the geometric method, the tilt factor and the shape factor of the reflecting surface. These were introduced for the first time, and the corresponding mathematical model was established. The method of numerical simulation is systematically studied and multimode optical fibre parameters (including optical fibre of axial spacing, optical fibre core diameter and numerical aperture) are adopted. The reflective surface inclination and shape factors on the RIM–FOS intensity modulation characteristics are studied according to the obtained light quasi-Gaussian distribution model, establishing a general three-intensity modulation function of fibre optic sensor. The results show that the intensity modulation characteristic of specular reflection is obviously better than that of the diffuse reflection surface, and the peak value of the modulation function is five times that of diffuse reflection. The intensity modulation characteristic decreases with increase in the roughness of the reflection surface. The system can not only complete the RIM–FOS characteristic simulation and characteristic testing functions, but can also start-up the test and not be affected by the ambient light interference and power fluctuation of the light source. The test stability is good with high repeatability.

#### Keywords

- Optical fibre sensor
- reflection intensity modulation
- mathematical model

#### AMS 2010 codes

- 30D99

According to the function of optical fibre, optical fibre sensors can be divided into two types: the sensing type, or functional fibre optic sensor; and the non-functional fibre optic sensor, or the optical fibre sensor. The optical fibre of the sensing fibre sensor not only transmits the light wave, but also senses the change in the measured parameters, that is, the acquisition and transmission of information are in the optical fibre, which has the characteristics of transmission and sensing. Optical fibre sensing technology is a new optical sensing technology with the development of optical fibre and its communication technology. The basic principle of optical fibre sensing detection is to use optical fibre in the light wave parameters (such as light intensity, frequency, wavelength, phase and polarisation state) with a change in the law of the measured parameters. By measuring these optical wave parameters to achieve the detection of external physical quantity, it is an important application of optical fibre in the field of non-communication. Reflective intensity modulation fibre-optic sensor has high sensitivity, is a simple structure and is not subject to electromagnetic interference. It can be used under high pressure, high temperature and harsh environment, and has the advantages of fast response speed, low cost and has been widely applied to the displacement, velocity, acceleration, pressure, temperature, flow, the electromagnetic field, the surface roughness measurement [1] and other physical quantities. For example, F. J. Arregui from Spain developed a new type of optical fibre sensor which can measure both the temperature and humidity at the same time. This shows that the optical fibre sensor plays a significant role in temperature and humidity measurement. Golnabi from Iran has presented the design and operation of a fibre optic sensor for mass measurement using a pair of fibre optic and reflective coated lenses whose light emphasis system is based on the relative motion of the lens. The design demonstrates the potential application of fibre optic sensors in mass measurement using the principle of intensity modulation. Hua-Yong Yang and others studied the geometric analysis method of single fibre of reflective optical fibre sensor theory model. They studied the intensity modulation function of the measured object surface inclination factor and the shape factor influence on the intensity modulation function shows that with increase in the tilt reflector and shape factors, the strength of the fibre optic sensor modulation function decreases, The measurement sensitivity is also reduced.

Intensity modulation function (IMF) is an important basis for detecting the position shift of reflective optical fibre sensors. The measurement results are easily affected by the inherent structure parameters of the fibre, the performance parameters of the fibre and the reflection characteristics of the reflecting surface. The light intensity modulation characteristic function model of the reflective intensity modulated fibre sensor is established in this paper through theoretical analysis. The influence of the mirror surface (roughness

The characteristic function curve of optical stress control is mainly determined by the ratio of the received light intensity value to the transmitted light intensity value and the distance between the reflected surface of the measured object and the end face of the optical fibre probe. The change in displacement can be calculated by a change in the value of the characteristic function of optical stress system. With applications in the actual measurement, the impact of optical fibre sensor light emphasises the function of many factors. Most of these factors are the inherent optical fibre sensor system, such as optical fibre sending and receiving optical fibre core diameter and numerical aperture, sending fibre and receiving optical fibre core centre distance, and the reflective surface properties and optical fiber probe end face to the object being measured the reflection surface distance. Therefore, the light stress control function can be expressed as a function related to the inherent factors of the system:

where _{1} and _{2} are the core radii of the transmitting fibre and the receiving fibre respectively; NA_{1} and NA_{2} are their numerical apertures respectively;

When the outgoing light intensity of the transmitting fibre is Gaussian, the variation of its light intensity with distance can be expressed as:

In the formula, _{t}

where _{N} = arcsin NA is the numerical aperture angle of the fibre, and the numerical aperture of the fibre. According to the Gaussian light intensity theory and Beckmann scattering theory, the light intensity model of Gaussian light reflected by the reflecting surface is as follows:

where

The optical power received by the receiving optical fibre is:

where _{R}

Among them,

To sum up, it can be concluded that with a change in the distance

Equation (10) is the optical stress control function of the optical fibre sensor when the output light intensity of the transmitting fibre is Gaussian distribution and the reflecting surface is a rough surface. According to Eq. (10), when the optical fibre probe of the optical fibre sensor is determined, the structural parameters and characteristic parameters of the optical fibre are fixed and unchanged. The intensity modulation function is only closely related to the distance

According to the light intensity distribution model of the outgoing end of the transmitting fibre obtained by diffraction analysis, and Beckmann’s theory of the scattering of light on reflective surfaces of different properties, we can give the reflected intensity distribution model of the quasi-Gaussian light passing through the reflecting surface, that is,

This is the general model of the light intensity reflected by a quasi-Gaussian beam emitted by a multi-mode transmitting fibre after passing through a reflective surface. When the reflecting surface is ideally smooth, due to

Therefore, in the case of ideal smooth reflection surface, the reflected light field and the outgoing light field of the transmitting fibre are essentially the same, and their distribution is also of quasi-Gaussian type, but there is a difference of reflectivity coefficient. In general, when the intensity modulation function of RIM–FOS is studied, the reflecting surface is usually determined as the mirror reflecting surface. In this case, the intensity modulation function of RIM–FOS is studied on the basis of Eq. (1). When RIM–FOS is used in industrial measuring applications and the reflecting surface is rough, Eq. (2) is used and parameters G and

The inner ring is composed of several TF clusters and the radius is _{1}, the core diameter of TF is _{1}, the outer ring radius is _{2} and the stem core diameter is _{2}. The numerical apertures of both TF and RF are Na. For a coaxial type I fibre bundle, the T in the centre can actually be considered as a coarse transmitting fibre with a radius of _{1} and a numerical aperture of NA. The thick TF and the outer layer are matched into optical fibre pairs one by one to calculate the combination of light intensity. Assume that the thickness of the cladding and adhesive is _{1}, and assume that the fabrication of the fibre bundle is so strict that the RF of the outer layer is completely arranged in the form of concentric circles, then it can be deduced that the RF of the outer layer can be arranged in

INT(•) represents a truncated round function (the same below). So we can calculate the i_{ri}

Therefore, on the premise that the optical fibre emphasises the control function _{pair}

To compare the intensity of the above two kinds of method of compensation effect, we adopt the spectroscope spectral compensation method and

As can be seen from the experimental results in Table 1, in the light intensity compensation experiment using a spectroscope, when the amplitude of the measured signal was nearly doubled due to the rotation of the polariser (from 0.6380 V to 1.1301 V), the ratio after compensation changed by about three times, indicating that the purpose of light intensity compensation could not be achieved at all in this case. However, in the intensity compensation experiment using ‘II’ fibre bundle, the output variation after compensation is still less than 3.0%, even when the measured intensity variation is nearly ten times. It can be seen that the ‘II’ type optical fibre bundle has a higher compensation accuracy compared with the spectroscopic method, and the former does not have harsh requirements on the polarisation characteristics of the incident light [8].

Experimental data of two kinds of spectral compensation

The town of spectroscopic Spectroscopic fill Their data | Measure the signal (v} | 0.6380 | 0.7075 | 0.8263 | 0.9326 | 0.9873 | 1.0646 | 1.1069 | 1.1241 | 1.1301 |

Reference signal (v} | 0.2554 | 0.3271 | 0.4896 | 0.6750 | 0.8049 | 1.0217 | 1.1940 | 1.2644 | 1.3470 | |

The ratio of | 2.498 | 2.163 | 1.688 | 1.382 | 1.227 | 1.042 | 0.927 | 0.889 | 0.839 | |

Type II fibre beam splitting compensation | Measure the signal (v} | 0.3776 | 0.91325 | 1.4256 | 2.0683 | 2.7413 | 3.2959 | 3.9183 | 4.3254 | 4.6840 |

Reference signal (v} | 0.2819 | 0.6692 | 1.5036 | 1.5437 | 20.569 | 2.4815 | 2.9666 | 3.2935 | 3.5949 | |

The ratio of | 1.339 | 1.365 | 1.353 | 1.340 | 1.333 | 1.328 | 1.321 | 1.313 | 1.303 |

The influence of specular reflection and diffuse reflection on the modulation function is analysed by changing the roughness

Figure 1 shows that the modulation function curve of the optical fibre sensor is significantly higher than that of the diffuse reflecting surface when the optical fibre sensor is reflected specular on the reflective surface, and the modulation function values are all higher than those of the diffuse reflecting surface, and the linear interval of the front and rear slopes and the sensitivity are better than those of the production function curve of the diffuse reflecting surface. When the reflection surface contains both specular reflection and diffuse reflection, the roughness

It can be seen from Fig. 2 that when the reflector roughness increases sequentially, the modulation function curve tends to decline, the peak value of the modulation function decreases sequentially, and the linear interval and the sensitivity of the front and rear slopes decrease with the increase of the reflector roughness. According to the simulation results, when the roughness of the reflecting surface is a smooth mirror, slightly rough surface, relatively rough surface and very rough surface, the intensity modulation characteristics of the optical fibre sensor decrease successively, and the linear interval used for measurement decreases, and the sensitivity decreases, which is not good for the actual measurement of the optical fibre sensor. As can be seen from the simulation results, on the premise that other parameters remain unchanged: (1) With the increase of core spacing, the signal light received by the receiving fibre decreases, and the light emphasis control function decreases accordingly; With the increase of the distance between the core and the shaft, the curve of the optical stress characteristic function decreases and moves backward, and the peak distance gradually increases with the increase of the distance between the core and the shaft. And the sensitivity of the foreslope gradually decreases [9,10]. From this analysis, it is concluded that the distance between the fibre core axis of the optical fibre probe should be as small as possible, that is, the sending fibre and the receiving fibre should be closely arranged within the range of conditions. The results show that the foreslope sensitivity and the peak modulation function of the intensity modulation characteristic curve of the fibre sensor are mainly affected by the core spacing of the fibre probe. Therefore, the sensor sensitivity can be improved by reducing the core spacing of the fibre probe in practical application.

Sent via theoretical analysis to establish when the optical fibre end light intensity of Gaussian distribution and reflective surface diffuse reflection occurs fibre optic sensor light emphasises the system function model. The intensity modulation characteristic analysis summarised in the design of the actual optical fibre sensor, in order to improve the precision of luminescent fibre core shaft spacing, shall be as far as possible in small, send the fibre core diameter should be reduced, The receiving fibre and the numerical aperture of the fibre should be as large as possible under the premise of meeting the measurement range. Different optical fibre parameters correspond to different intensity modulation characteristics. In the practical application of reflective intensity modulation optical fibre sensor, appropriate optical fibre parameters should be selected according to the specific requirements to achieve the ideal measurement requirements. In this paper, the research and application progress of rim-fos since the invention of the first patent for the application of reflective strength optical fibre sensor are reviewed comprehensively, and the important achievements in the research of this kind of optical fibre sensor in the past 40 years as well as the existing theoretical and technical problems are pointed out. For the basic structure of RIM-FOS fiber pair, the tilt factor and shape factor of the reflector are introduced for the first time in this paper. The optical fibre parameters (including the spacing

#### Experimental data of two kinds of spectral compensation

The town of spectroscopic Spectroscopic fill Their data | Measure the signal (v} | 0.6380 | 0.7075 | 0.8263 | 0.9326 | 0.9873 | 1.0646 | 1.1069 | 1.1241 | 1.1301 |

Reference signal (v} | 0.2554 | 0.3271 | 0.4896 | 0.6750 | 0.8049 | 1.0217 | 1.1940 | 1.2644 | 1.3470 | |

The ratio of | 2.498 | 2.163 | 1.688 | 1.382 | 1.227 | 1.042 | 0.927 | 0.889 | 0.839 | |

Type II fibre beam splitting compensation | Measure the signal (v} | 0.3776 | 0.91325 | 1.4256 | 2.0683 | 2.7413 | 3.2959 | 3.9183 | 4.3254 | 4.6840 |

Reference signal (v} | 0.2819 | 0.6692 | 1.5036 | 1.5437 | 20.569 | 2.4815 | 2.9666 | 3.2935 | 3.5949 | |

The ratio of | 1.339 | 1.365 | 1.353 | 1.340 | 1.333 | 1.328 | 1.321 | 1.313 | 1.303 |

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