Matrix method for polarization transmission analysis of polarization maintaining optical fiber

Abstract

The invention relates to the technical field of polarization optics, and discloses a matrix method for polarization transmission analysis of a polarization-maintaining optical fiber, which comprises the following steps: S1, establishing a polarization angle measurement system based on the polarization-maintaining optical fiber, the polarization angle measurement system comprises an upper instrument and a lower instrument, and the upper instrument and the lower instrument are connected through the polarization-maintaining optical fiber; the upper instrument is provided with a polarization state generator, and the lower instrument is provided with a polarization state analyzer; s2, establishing a matrix model of a polarization maintaining optical fiber through the polarization angle measurement system in the S1, wherein the matrix model of the polarization maintaining optical fiber comprises a matrix model of a temperature effect, a matrix model of optical fiber bending, a matrix model of a magnetic field effect and a matrix model of a coupling effect; and S3, simulation analysis: simulating the polarization transmission process of the polarization maintaining optical fiber under different environmental factors, and verifying the correctness of the matrix model derivation of the polarization maintaining optical fiber in the S2. Based on the Jones matrix, the characteristic matrix of the polarization-maintaining optical fiber is realized, and the analysis complexity of the polarization-maintaining optical fiber in a complex system is simplified.

Description

Technical Field

[0001] The present invention relates to the technical field of polarization optics, and particularly relates to a matrix method for analyzing the polarization transmission of polarization-maintaining fibers. Background Art

[0002] In the field of polarization optics, polarization-maintaining fibers have become indispensable key components due to their excellent polarization-maintaining ability, especially playing an important role in systems that require precise angle measurement. Such systems rely on the unique vibration direction characteristics of linearly polarized light to achieve high-precision angle measurement, and polarization-maintaining fibers, with their unique stress-induced high birefringence characteristics, ensure the stable transmission of the polarization state, thereby enabling angle measurement under long-distance and non-line-of-sight conditions.

[0003] However, during the actual application process of polarization-maintaining fibers, their performance is extremely vulnerable to changes in the external environment, especially factors such as temperature, fiber bending, and magnetic fields. These external factors change the stress state inside the fiber, thereby affecting the birefringence characteristics of the fiber, resulting in changes in the polarization state and a decrease in measurement accuracy. Existing research shows that although polarization-maintaining fibers have good polarization-maintaining ability, their sensitivity to environmental temperature, stress changes caused by bending, and characteristic drift under the action of magnetic fields cannot be ignored.

[0004] To address these challenges, scholars have proposed various methods to study and analyze the behavior characteristics of polarization-maintaining fibers in different environments. For example, using a distributed polarization analyzer to accurately measure the birefringence varying with temperature, developing multi-parameter sensors to simultaneously monitor temperature and strain, and reducing the impact of temperature fluctuations by adjusting the geometric effect of the fiber core. At the same time, some scholars have also deeply explored the effects of torsion, electric fields, and magnetic fields on the performance of polarization-maintaining fibers through simulation and experimental means.

[0005] Although these studies provide important references for understanding the performance of polarization-maintaining fibers, most of them focus on the influence of single components of the fiber or single environmental factors, and it is difficult to comprehensively reflect the complex situation of the interaction between components and the comprehensive action of environmental factors in an angle measurement system with polarization. Summary of the Invention

[0006] Aiming at the above deficiencies in the prior art, the matrix method for analyzing the polarization transmission of polarization-maintaining fibers provided by the present invention solves the problem that the angle measurement system of polarization-maintaining fibers in the prior art is only applicable to the study of single components of polarization-maintaining fibers and is difficult to achieve the overall study of complex systems.

[0007] To achieve the above invention objective, the technical solution adopted by the present invention is: A matrix method for analyzing the polarization transmission of polarization-maintaining fibers, comprising the following steps:

[0008] S1. Establish a polarization angle measurement system based on polarization-maintaining fiber. The polarization angle measurement system includes an upper instrument and a lower instrument, which are connected by polarization-maintaining fiber. A polarization state generator is set on the upper instrument, and a polarization state analyzer is set on the lower instrument.

[0009] S2. Establish a matrix model of the polarization-maintaining fiber through the polarization angle measurement system in S1. The matrix model of the polarization-maintaining fiber includes a matrix model of temperature effect, a matrix model of fiber bending, a matrix model of magnetic field effect, and a coupling effect matrix model.

[0010] S3. Simulation analysis: Simulate the polarization transmission process of the polarization-maintaining fiber under different environmental factors to verify the correctness of the derivation of the matrix model of the polarization-maintaining fiber in S2.

[0011] Furthermore, for the matrix method for analyzing the polarization transmission of the polarization-maintaining fiber, S2 is specifically as follows:

[0012] S21: After the incident light emitted from the upper instrument and passing through the polarization state generator passes through the polarization-maintaining fiber, the emerging polarized light is obtained. The matrix relationship between the Jones vector E0 of the incident light and the Jones vector E1 of the emerging light is given by equations (1) and (2):

[0013]

[0014]

[0015] where ω is the transmission frequency related to the wavelength of the transmitted polarized light, ω = 2πc / λ, c is the speed of light, λ is the wavelength of the transmitted polarized light, δ is the phase difference between the modes of the polarization-maintaining fiber, δ = δ0 + Δδ, which is divided into the inherent phase difference δ0 and the variable phase difference Δδ;

[0016] The inherent phase difference δ0 is affected by the wavelength λ of the transmitted polarized light, the inherent birefringence difference Δn, and the length L of the polarization-maintaining fiber as shown in equation (3):

[0017]

[0018] S22. Establish a matrix model of temperature effect: When the polarization-maintaining fiber is under the action of temperature, the relationship between the phase difference and the temperature change amount ΔT, the temperature effect length l T and the incident light wavelength λ is given by equation (4):

[0019]

[0020] S23. Establish a matrix model of fiber bending: When the polarization-maintaining fiber is bent, the phase difference is affected by the magnitude of the curvature radius R, the fiber bending length l C and the incident wavelength λ as shown in equation (5):

[0021]

[0022] S24. Establish a matrix model of the magnetic field effect: When the polarization-maintaining fiber is affected by a magnetic field, the phase difference is affected by magnetic field parameters such as the magnetic field quantity H along the fiber axis component and the magnetic field action length l M The relationship is shown in Equation (6):

[0023]

[0024] S25. Establish a coupling effect matrix model: When the polarization-maintaining fiber is affected by temperature, magnetic field, and fiber bending, the change in the phase difference is shown in Equation (7):

[0025]

[0026] Furthermore, for the matrix method of the polarization transmission analysis of the polarization-maintaining fiber, S22 is specifically as follows:

[0027] S221. Under the action of temperature on the polarization-maintaining fiber, the birefringence change in the polarization-maintaining fiber is shown in Equation (8):

[0028]

[0029] Among them, Δα is the difference between the stress area and the core thermal expansion coefficient of the panda-type polarization-maintaining fiber, E is Young's modulus, C is the photoelastic coefficient, ε is the ellipticity of the core of the polarization-maintaining fiber ε = a / b, a and b are the core radii of the slow and fast axes respectively, υ is the Poisson's ratio coefficient of the panda-type polarization-maintaining fiber, ΔT is the temperature change, n x is the refractive index of the slow axis (X-axis), n y is the refractive index of the fast axis (Y-axis);

[0030] S222. When the temperature effect length is l T , and the incident light wavelength is λ, the polarization-maintaining fiber in the temperature effect will cause a phase difference between the two modes as shown in Equation (9):

[0031]

[0032] S223. Substitute Equation (8) into Equation (9) to obtain the influence of temperature on the phase difference as shown in Equation (10):

[0033]

[0034] S224. Substitute Equation (10) into Equation (2) to obtain the relationship between the phase difference and the temperature change ΔT, the temperature effect length l T and the incident light wavelength λ, and further obtain the matrix model of the temperature effect as shown in Equation (4):

[0035]

[0036] Furthermore, for the matrix method for analyzing the polarization transmission of polarization-maintaining optical fiber, S23 is specifically as follows:

[0037] S231. When the polarization-maintaining optical fiber is bent, the birefringence difference is given by Equation (11):

[0038]

[0039] where: n is the refractive index of the core material, υ is the Poisson's ratio coefficient, p2 and p1 are the photoelastic tensors, A is the outer diameter of the polarization-maintaining optical fiber, and R is the magnitude of the radius of curvature;

[0040] S232. When the length of the bent part of the optical fiber is l C , and the wavelength of the incident light is λ, the phase difference between the two modes caused by the bending effect of the optical fiber is given by Equation (12):

[0041]

[0042] Substituting Equation (12) into Equation (11), the influence of the optical fiber bending on the phase difference is obtained as Equation (13):

[0043]

[0044] Substituting Equation (13) into Equation (2), the relationship between the phase difference and the magnitude of the radius of curvature R, the length l of the optical fiber bending C and the incident wavelength λ is obtained as Equation (5):

[0045]

[0046] Furthermore, for the matrix method for analyzing the polarization transmission of polarization-maintaining optical fiber, S24 is specifically as follows:

[0047] S241. When the polarization-maintaining optical fiber is affected by a magnetic field, the birefringence difference of the polarization-maintaining optical fiber is given by Equation (14):

[0048]

[0049] where λ is the wavelength of the incident light, V is the Verdet constant, and H is the magnetic field quantity of the axial component of the optical fiber;

[0050] S242. When the length of the magnetic field is l M , the phase difference between the two modes of the polarization-maintaining optical fiber under the action of the magnetic field is given by Equation (15):

[0051]

[0052] S243. Substituting Equation (14) into Equation (15), the influence of the magnetic field on the phase difference is obtained as Equation (16):

[0053] Δδ M = 2VHl M (16);

[0054] S244. Substitute Equation (16) into Equation (2), and the relationship between the phase difference and the magnetic field parameters such as the magnetic field quantity H along the axial direction of the optical fiber and the magnetic field action length l M is Equation (6):

[0055]

[0056] Furthermore, the matrix method for the polarization transmission analysis of polarization-maintaining optical fiber in the above is specifically as follows in S25:

[0057] S251. The length L of the polarization-maintaining optical fiber is decomposed into countless Δl, and the common influence factor is regarded as the individual action under each small segment Δl. The relationship between each component is Equation (17):

[0058] J = J n …J3J2J1(17);

[0059] S252. Expand each part of the matrix component to obtain Equation (18):

[0060]

[0061] S253. Perform mathematical calculations on Equation (18) to obtain Equation (19):

[0062]

[0063] S254. By combining the influencing factors of temperature, magnetic field and optical fiber bending, Equation (20) is obtained:

[0064]

[0065] S255. When the polarization-maintaining optical fiber is affected by temperature, magnetic field and optical fiber bending, the change in the phase difference is Equation (7):

[0066]

[0067] The beneficial effects of the present invention are as follows: Based on the Jones matrix theory, the present invention represents each component in the system through a matrix, constructs a unified matrix framework to comprehensively analyze the comprehensive influence of the parameters of each component and environmental factors on the overall performance in a complex system. The present invention introduces the environmental factors (temperature effect, fiber bending, and magnetic field effect) that affect the transmission performance of polarization-maintaining fiber and the characteristic parameters of the polarization-maintaining fiber itself into the matrix model, and constructs a more comprehensive matrix model with these factors as matrix parameters. The derivation and simulation research of the matrix model show that when linearly polarized light passes through the polarization-maintaining fiber under different environmental factors, the phases of the two modes on the fast axis and the slow axis will change, and ultimately the polarization state of the outgoing light changes periodically with the parameters of the environmental factors. The present invention provides a new basis and technical support for improving the overall accuracy and stability of the angle measurement system with polarization. Description of the Drawings

[0068] Figure 1 It is a schematic structural diagram of a polarization angle measurement system based on polarization-maintaining fiber;

[0069] Figure 2 It is a schematic diagram of the influence of environmental factors on polarization-maintaining fiber;

[0070] Figure 3 It is a schematic diagram of polarization-maintaining fiber under temperature effect;

[0071] Figure 4 It is a schematic diagram of the bending of polarization-maintaining fiber;

[0072] Figure 5 It is a schematic diagram of polarization-maintaining fiber when affected by a magnetic field;

[0073] Figure 6 It is a differential schematic diagram of polarization-maintaining fiber under multiple environments;

[0074] Figure 7 It is a schematic diagram of a bent polarization-maintaining fiber under a magnetic field;

[0075] Figure 8 It is a schematic diagram of E changing with l T when ΔT is fixed;

[0076] Figure 9 It is a schematic diagram of E changing with ΔT T when l is fixed;

[0077] Figure 10 It is a schematic diagram of E changing with ΔT and l T ;

[0078] Figure 11 It is a schematic diagram of E changing with T E ;

[0079] Figure 12is the schematic diagram of the variation of E with l when R is fixed C ;

[0080] Figure 13 is the schematic diagram of the variation of E with R when l is fixed C ;

[0081] Figure 14 is the schematic diagram of the variation of E with R and l C ;

[0082] Figure 15 is the schematic diagram of the variation of E with R E ;

[0083] Figure 16 is the variation of E with H when l is fixed M ;

[0084] Figure 17 is the variation of E with l when H is fixed M ;

[0085] Figure 18 is the schematic diagram of the variation of E with H and l M ;

[0086] Figure 19 is the schematic diagram of the variation of E with H E ; Specific Embodiments

[0087] The following describes the specific embodiments of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions made using the concept of the present invention are within the scope of protection.

[0088] This embodiment provides a matrix method for analyzing the polarization transmission of polarization-maintaining optical fibers, as shown Figure 1 in the figure, which shows a polarization angle measurement system based on a polarization-maintaining optical fiber. The transmission between the upper instrument and the lower instrument is carried out using a polarization-maintaining optical fiber. The laser beam in the upper instrument is converted into linearly polarized light by a polarization state generator, and the linearly polarized light carrying angle information is coupled into the polarization-maintaining optical fiber and transmitted to the magneto-optical modulation system of the lower instrument. The polarization light signal is modulated by the generated alternating magnetic field. The modulated polarization light completes beam expansion through a beam expansion system. Then, the polarization light passes through a polarization state analysis system, photoelectric conversion, signal detection and processing. The electrical signal corresponding to the angle is extracted to complete the angle measurement.

[0089] As shown Figure 2As shown in the figure, it is a schematic diagram of the influence of environmental factors on polarization-maintaining fiber. The incident light is linearly polarized, and its polarization state can be represented by the Jones vector $E_0 = \begin{bmatrix} E \\ E \end{bmatrix} \cdot e^{j\omega t}$. And after passing through the polarization-maintaining fiber, the output light is polarized, and its polarization state can be represented by the Jones vector $E_1 = \begin{bmatrix} E \\ E \end{bmatrix}$. Due to its high birefringence characteristics, the polarization-maintaining fiber divides the polarized light into two orthogonal modes. Since the propagation constants between the two orthogonal modes are different, a phase difference $\delta$ is generated between these two modes. When the polarization-maintaining fiber is affected by environmental factors such as temperature, fiber bending, or magnetic field, the internal stress distribution and refractive index distribution will change, which will cause fluctuations in the phase difference $\delta$ between the two orthogonal propagation modes. Based on Fresnel's law, the change in the phase difference $\delta$ of the polarization-maintaining fiber can be represented by a Jones matrix. The matrix relationship between the Jones vector $E_0$ of the incident light and the Jones vector $E_1$ of the output light is: x0 E y0 T ·e iωt As shown in the figure, it is a schematic diagram of the influence of environmental factors on polarization-maintaining fiber. The incident light is linearly polarized, and its polarization state can be represented by the Jones vector $E_0 = \begin{bmatrix} E \\ E \end{bmatrix} \cdot e^{j\omega t}$. And after passing through the polarization-maintaining fiber, the output light is polarized, and its polarization state can be represented by the Jones vector $E_1 = \begin{bmatrix} E \\ E \end{bmatrix}$. Due to its high birefringence characteristics, the polarization-maintaining fiber divides the polarized light into two orthogonal modes. Since the propagation constants between the two orthogonal modes are different, a phase difference $\delta$ is generated between these two modes. When the polarization-maintaining fiber is affected by environmental factors such as temperature, fiber bending, or magnetic field, the internal stress distribution and refractive index distribution will change, which will cause fluctuations in the phase difference $\delta$ between the two orthogonal propagation modes. Based on Fresnel's law, the change in the phase difference $\delta$ of the polarization-maintaining fiber can be represented by a Jones matrix. The matrix relationship between the Jones vector $E_0$ of the incident light and the Jones vector $E_1$ of the output light is: x1 E y1 T Due to its high birefringence characteristics, the polarization-maintaining fiber divides the polarized light into two orthogonal modes. Since the propagation constants between the two orthogonal modes are different, a phase difference $\delta$ is generated between these two modes. When the polarization-maintaining fiber is affected by environmental factors such as temperature, fiber bending, or magnetic field, the internal stress distribution and refractive index distribution will change, which will cause fluctuations in the phase difference $\delta$ between the two orthogonal propagation modes. Based on Fresnel's law, the change in the phase difference $\delta$ of the polarization-maintaining fiber can be represented by a Jones matrix. The matrix relationship between the Jones vector $E_0$ of the incident light and the Jones vector $E_1$ of the output light is:

[0090]

[0091] where $\omega$ is the propagation frequency related to the wavelength of the transmitted polarized light, $\omega = 2\pi c / \lambda$, $c$ is the speed of light, $\lambda$ is the wavelength of the transmitted polarized light, and $\delta$ is the phase difference between the modes of the polarization-maintaining fiber, $\delta = \delta_0 + \Delta\delta$, which is divided into the inherent phase difference $\delta_0$ and the changing phase difference $\Delta\delta$. The inherent phase difference $\delta_0$ is affected by the wavelength $\lambda$ of the transmitted polarized light, the inherent birefringence difference $\Delta n$, and the length $L$ of the polarization-maintaining fiber.

[0092]

[0093] The changing phase difference $\Delta\delta$ is affected by typical environmental factors such as temperature $T$, fiber bending $R$, and magnetic field $H$. The following gives the theoretical derivations of the individual effects and coupled effects of each environmental factor.

[0094] As Figure 3 shown in the figure, it is a schematic diagram of the polarization-maintaining fiber under the temperature effect. The length of the polarization-maintaining fiber is $L$, and the length of the temperature effect is $l$. T ​​。A polarization-maintaining fiber is a crystalline material with a cladding wrapped around the core. When the temperature changes, the birefringence in the polarization-maintaining fiber is affected by the thermal expansion effect, the radial stress thermal effect, and the axial strain thermal effect. Therefore, when the temperature changes, the thermal expansion coefficients of the cladding and the core are different, and the radial and axial thermal effects will cause additional strain in the polarization-maintaining fiber, thereby changing the birefringence of the polarization-maintaining fiber. The relationship between the thermal effect and the birefringence of different types of polarization-maintaining fibers is different. In this paper, taking the panda-type polarization-maintaining fiber as an example, temperature-related parameters are introduced into the matrix for characterization. The birefringence of the panda-type polarization-maintaining fiber mainly comes from two stress regions generated by the doped quartz rods on the panda eyes. Therefore, the change in birefringence caused by the stress region under temperature change is mainly discussed. According to the thermo-optic effect and the photoelastic effect, there is birefringence in the panda-type polarization-maintaining fiber under the action of temperature:

[0095] where Δα is the difference between the thermal expansion coefficients of the stress region and the core of the panda-type polarization-maintaining fiber, E is the Young's modulus, C is the photoelastic coefficient, ε is the ellipticity of the core of the polarization-maintaining fiber ε = a / b, a and b are the radii of the fast and slow axes of the core respectively, υ is the Poisson's ratio coefficient of the panda-type polarization-maintaining fiber, ΔT is the temperature change, n x is the refractive index of the slow axis (X-axis), n y is the refractive index of the fast axis (Y-axis). When the temperature effect length is l T , and the incident light wavelength is λ, the polarization-maintaining fiber in the temperature effect will cause a phase difference between the two modes as:

[0096] Substituting Equation (4) into Equation (5), the influence of temperature on the phase difference is obtained as:

[0097]

[0098] Substituting Equation (10) into Equation (2), the relationship between the phase difference and the temperature change ΔT, the temperature effect length l T and the incident light wavelength λ can be obtained, and then the matrix model of the panda-type polarization-maintaining fiber is obtained as:

[0099]

[0100] According to the derivation process, this matrix model can be extended to different types of polarization-maintaining fibers by replacing the temperature relationship equation in the panda-type polarization-maintaining fiber.

[0101] As Figure 4 shown, it is a schematic diagram of the polarization-maintaining fiber when it is bent. The bending length of the polarization-maintaining fiber is l C, the radius of curvature R represents the degree of bending. Once the polarization-maintaining fiber is made, its shape and stress value are fixed. However, when the fiber is bent, extrusion occurs, generating an external force that causes induced birefringence and changes the birefringence difference through the photoelastic effect. According to the photoelastic effect, the birefringence difference caused by pure bending of the polarization-maintaining fiber is:

[0102]

[0103] In the panda-type polarization-maintaining fiber, let n be the refractive index of the core material, υ be the Poisson's ratio coefficient, p2 and p1 be the photoelastic tensors, A be the outer diameter of the polarization-maintaining fiber, and R be the magnitude of the radius of curvature. When the length of the bent part of the fiber is l C , and the wavelength of the incident light is λ, the phase difference between the two modes caused by the bending effect of the fiber is:

[0104] Substituting Equation (12) into Equation (11), the influence of fiber bending on the phase difference is obtained as:

[0105]

[0106] Substituting Equation (13) into Equation (2), we can obtain that the phase difference is affected by parameters such as the magnitude of the radius of curvature R, the length l of fiber bending C and the incident wavelength λ, and the matrix model of the panda-type polarization-maintaining fiber is:

[0107]

[0108] As Figure 5 shown, it is a schematic diagram of the polarization-maintaining fiber under the influence of a magnetic field. The length of the polarization-maintaining fiber is L, and the length of the magnetic field is l M . The core of the polarization-maintaining fiber is a crystal medium for transmitting polarized light. According to the Faraday effect, it is known that the axial component along the polarization-maintaining fiber will cause a change in the dielectric constant of the medium in a magnetic field environment, and the change in the dielectric constant will cause the rotation effect of polarized light in the polarization-maintaining fiber and change the polarization direction. In this paper, taking the panda-type polarization-maintaining fiber as an example, the influence parameters of the magnetic field are introduced into the matrix of the polarization-maintaining fiber for characterization. According to the Faraday effect, the birefringence difference of the panda-type polarization-maintaining fiber under the action of a magnetic field is:

[0109]

[0110] where λ is the wavelength of the incident light, V is the Verdet constant, and H is the magnetic field quantity of the axial component of the fiber. When the length of the magnetic field is l M , the phase difference between the two modes of the polarization-maintaining fiber under the action of a magnetic field is:

[0111]

[0112] Substituting Equation (14) into Equation (15), the influence of the magnetic field on the phase difference is obtained as follows:

[0113] Δδ M =2VHl M (16)

[0114] Substituting Equation (16) into Equation (2), we can obtain that the phase difference is affected by magnetic field parameters such as the magnetic field quantity H along the fiber axial direction and the magnetic field action length l M and the matrix model of the panda polarization-maintaining fiber is:

[0115]

[0116] Through the above theoretical derivation, we systematically obtained the matrix model of the influence of each environmental factor on the polarization-maintaining fiber. The application of polarization-maintaining fibers often faces the multi-factor action of the environment in a complex system. These composite environmental factors jointly affect the polarization-maintaining fiber, resulting in fluctuations in the birefringence characteristics, and further causing significant changes in the phase difference of the transmitted polarized light. By analyzing the previous equations, it is known that the influence parameters of the three environmental factors on birefringence are different and interact with each other. Therefore, the coupling relationship between the phase differences is deduced as Figure 6 .

[0117] When the polarization-maintaining fiber is affected by multiple environmental factors simultaneously, we decompose the fiber into the differential form as Figure 6 shown. The length L of the polarization-maintaining fiber is decomposed into countless Δl, and the common influence factor is regarded as the individual action under each small segment Δl. Therefore, according to the Jones matrix principle, the relationship between the components can be obtained as:

[0118] J=J n …J3J2J1(17)

[0119] Expand each part of the component with the matrix

[0120]

[0121] Performing mathematical calculations on Equation (18), we get:

[0122]

[0123] Equation (19) is obtained by combining each term in the temperature, magnetic field, and fiber bending factors:

[0124]

[0125] where δ T is the combination of all differential lengths under the influence of temperature, δ M is the combination of all differential lengths under the influence of the magnetic field, and δ CIt is the combination of all differential lengths under the influence of fiber bending.

[0126]

[0127] According to the derivation, it can be seen that the coupling effect of polarization-maintaining fiber can be obtained by multiplying the matrices of various environmental factors. The special analysis of the coupling of fiber bending and magnetic field effect is in Figure 7 When the magnetic field and the fiber bending factor are coupled, it can be found that part of the magnetic field is not along the direction of the axis of the polarization-maintaining fiber. Therefore, the following analysis is required.

[0128] As Figure 7 shown, it is a bent polarization-maintaining fiber under a magnetic field. Project the magnetic field shown in the figure onto the axial and perpendicular axes.

[0129] H || = Hcosθ (21)

[0130] H ⊥ = Hsinθ(22)

[0131] Then, substitute equations (20) and (21) into equation (16):

[0132] Δδ M = 2V(Hcosθ)Δl M (23)

[0133] Performing mathematical transformation on the above equation, we can obtain:

[0134] Δδ M = 2VH(Δl M cosθ) = 2VHΔl rM (24)

[0135] Therefore, l M cosθ represents the projection of the length in the direction of the magnetic field. It can be concluded that when the fiber bending is coupled with the magnetic field, the actual length calculated under the magnetic field should be the length lr in the direction of the magnetic field M .

[0136] According to the derived temperature effect matrix model, when the incident light E0 = [E x0 E y0 T ·e iωt When transmitted through the polarization-maintaining fiber, the output light is:

[0137]

[0138] The polarization mode components on the X-axis and Y-axis are combined to obtain the polarization photoelectric field vector amplitude E at any instant:

[0139]

[0140] According to the analysis of formula (25), the phases of the two modes on the fast axis and the slow axis are affected by the temperature change ΔT and the temperature effect length l T . According to the transformation of Euler's formula, the phase difference between the two modes on the fast axis and the slow axis changes periodically with ΔT and l T . Therefore, the amplitude E of the polarized optical electric field vector after the combination of the two modes in formula (26) also shows periodic changes.

[0141] As Figure 8 shown, when the temperature change ΔT is a fixed value, taking ΔT = 5 °C (the first picture), ΔT = 10 °C (the second picture), and ΔT = 20 °C (the third picture) respectively, E changes periodically with l T , and the period value is related to the temperature change ΔT.

[0142] As Figure 9 shown, when the temperature action length l T is a fixed value, taking l T = 0.1 m (the first picture), l T = 0.3 m (the second picture), and l T = 0.5 m (the third picture) respectively, the change of E with ΔT also shows periodicity, and the period value is related to the length l T .

[0143] When both ΔT and l T change, the value range of ΔT is 0 to 30 °C, and the value range of l T is 0 to 1 m. As Figure 10 shown, the amplitude of the electric field vector of the outgoing polarized light shows a variable-period sine form that changes with ΔT or l T , and when ΔT and l T are small, the period is larger, and the larger the period means the smaller the influence. Because according to formula (25), the two modes exist in the form of trigonometric functions. The two variables ΔT and l T are mapped onto the traditional cosine function cos(ω t ). When one of them is used as the independent variable t, the other is mapped onto the frequency ω, thus showing a variable period.

[0144] As Figure 10 shown, set TE = ΔT·l T , and define the degree of the temperature effect through the temperature change ΔT and the temperature action length l T . Then, the change of E with the temperature influence degree TE shows periodicity, and this periodicity is related to the wavelength of the incident light. The frequency is ω = 2π·Δα·E·C·ε·λ-1 ·(1 + υ) -1 and the period value is T = λ·(1 + υ)·(Δα·E·C·ε) -1 . When the incident light is determined, the period is a constant. As Figure 11 shown.

[0145] According to the derived fiber bending matrix model, when the incident light E0 = [E x0 E y0 T ·e iωt passes through the polarization-maintaining fiber, the outgoing light is:[[]]

[0146]

[0147] According to the analysis of Equation (27), the phases of the two modes along the fast and slow axes are affected by the curvature radius R and the bending length l C . Combining the two modes in the X-axis and Y-axis directions according to Equation (26), the polarization optical electric field vector amplitude E at any time is obtained. According to the transformation of Euler's formula, we know that E changes periodically with the curvature radius R and the bending length l C . As Figure 12 shown, when the curvature radius R is a constant value, take R = 0.1 m (the first picture), R = 0.15 m (the second picture) and R = 0.2 m (the third picture) respectively. E changes periodically with the bending length l C , and the period value is related to R.

[0148] As Figure 13 shown, when the curvature radius R is a constant value, take l C = 0.1 m (the first picture), l C = 0.3 m (the second picture) and l C = 0.5 m (the third picture) respectively. E changes in a non-periodic form with the change of the curvature radius R. E shows an oscillating form with R because the denominator of Equation (27) contains the square of R, and when R is small, the oscillation of E is faster.

[0149] As Figure 14 shown, when R and l C change simultaneously, the value range of R is 0.03 - 0.5 m (because when R = 0, the frequency is too high to be observed), and the value range of l C is 0 - 1 m. As Figure 14 shown, E changes in a non-periodic manner with the change of the curvature radius R and the bending length l C . When l C ​When it is small, the period is large; when R is large, the period is also large, and a larger period indicates a smaller influence. When R > 0.5 m, the influence of fiber bending on polarization-maintaining fiber is small.

[0150] As Figure 15 shown, define R E = l C / R 2 as the degree of bending effect, which is affected by the curvature radius R and the fiber bending length l C . Then, E varies periodically with the degree of bending effect R E , which is related to the wavelength of the incident light. The frequency is ω = 0.5π·λ -1 ·n 3 ·(1 + υ)·(p2 - p1)·A 2 , and the period value is T = (4λ) / (n 3 ·(1 + υ)·(p2 - p1)·A 2 ). When the wavelength of the incident light is determined, the period value is a constant.

[0151] According to the derived magnetic field effect matrix model, when the incident light E0 = [E x0 E y0 T ·e iωt propagates through the optical fiber, the output light is:

[0152]

[0153] It can be inferred from the above formula (28) that the phases of the two modes along the fast axis and the slow axis are affected by the magnetic field strength H and the magnetic field action length l M . According to formula (26), the two modes on the X-axis and Y-axis are combined to obtain the polarization optical electric field vector amplitude E at any time. According to Euler transformation, E varies periodically with the magnetic field strength H and the magnetic field length l M .

[0154] As Figure 17 shown, when the magnetic field strength H is respectively the constant H = 0.5·10 6 A / m (the first picture), H = 1·10 6 A / m (the second picture) and H = 2·10 6 A / m (the third picture), E varies periodically with the magnetic field action length l M , and the period value is related to the value of H.

[0155] As Figure 16 shown, when the degree of magnetic field effect l M is a constant value, which are respectively l M = 0.3 m (the first picture), lM = 0.5 m (Second picture), l M = 1 m (Third picture). E exhibits variable periodicity with the magnetic field strength H.

[0156] As Figure 18 shown, according to the law, H E = H·l M is defined as the degree of magnetic field effect, which is affected by the magnetic field strength H and the magnetic field effect length l M . Then, E exhibits periodicity with the degree of magnetic field effect H E with a frequency of ω = 2V, a period value of T = π / V, and the period is a constant, as Figure 19 shown.

[0157] Based on the Jones matrix, this paper realizes the characteristic matrix of polarization-maintaining fiber, simplifies the analysis complexity of polarization-maintaining fiber in complex systems. Further, environmental factors (temperature effect, fiber bending, and magnetic field effect) that significantly affect the transmission performance of polarization-maintaining fiber and the characteristic parameters of polarization-maintaining fiber itself are introduced into the matrix model, and these factors are used as matrix parameters to construct a more comprehensive matrix model. The derivation and simulation study of the matrix model show that when linearly polarized light passes through polarization-maintaining fiber under different environmental factors, the phases of the two modes on its fast axis and slow axis will change, ultimately resulting in the polarization state of the output light changing periodically with the parameters of environmental factors. For the temperature effect, the change amount mainly depends on the temperature effect amount ΔT and the temperature effect length l T , and their product is defined as the temperature effect degree T E . When linearly polarized light passes through polarization-maintaining fiber, its polarization state changes periodically with the temperature effect degree, and the period is 5.24. For bent polarization-maintaining fiber, the combination of the bending length l C and the radius of curvature R is defined as the bending degree R E . The polarization state of the transmitted light changes periodically with the bending effect degree, and the period is 249.74. In a magnetic field environment, the product of the magnetic field quantity H and the magnetic field effect length l M is defined as the magnetic field effect degree H E , and the transmitted light changes periodically with the magnetic field effect degree. In this paper, we use the matrix model of polarization-maintaining fiber to quantify the influence of environmental factors on the transmission performance of polarization-maintaining fiber in complex systems. This research provides a more efficient and intuitive matrix model analysis method.

Claims

1. A matrix method for polarization transmission analysis of polarization-maintaining optical fibers, characterized in that It includes the following steps: S1. Establish a polarization angle measurement system based on polarization-maintaining fiber. The polarization angle measurement system includes an upper instrument and a lower instrument, and the upper instrument and the lower instrument are connected by polarization-maintaining fiber. A polarization state generator is arranged on the upper instrument, and a polarization state analyzer is arranged on the lower instrument; S2. Establish a matrix model of the polarization-maintaining fiber through the polarization angle measurement system in S1. The matrix model of the polarization-maintaining fiber includes a matrix model of temperature effect, a matrix model of fiber bending, a matrix model of magnetic field effect, and a coupling effect matrix model; S3. Conduct simulation analysis to simulate the polarization transmission process of the polarization-maintaining fiber under different environmental factors, and verify the correctness of the derivation of the matrix model of the polarization-maintaining fiber in S2.

2. The matrix method for polarization transmission analysis of polarization-maintaining optical fiber according to claim 1, characterized in that The specific content of S2 is as follows: S21: After the incident light emitted from the upper instrument and passing through the polarization state generator passes through the polarization-maintaining fiber, the outgoing polarized light is obtained. The matrix relationship between the Jones vector E0 of the incident light and the Jones vector E1 of the outgoing light is shown in Equation (1) and Equation (2): Where ω is the transmission frequency related to the wavelength of the transmitted polarized light, ω = 2πc / λ, c is the speed of light, λ is the wavelength of the transmitted polarized light, δ is the phase difference between the modes of the polarization-maintaining fiber, δ = δ0 + Δδ, which is divided into the inherent phase difference δ0 and the variable phase difference Δδ; The influence of the inherent phase difference δ0 on the wavelength λ of the transmitted polarized light, the inherent birefringence difference Δn, and the length L of the polarization-maintaining fiber is shown in Equation (3): S22. Establish a matrix model of the temperature effect: When the polarization-maintaining optical fiber is under the action of temperature, the relationship between the phase difference and the temperature change ΔT, the temperature effect length l T and the incident light wavelength λ is given by Equation (4): S23. Establish a matrix model for fiber bending: When the polarization-maintaining fiber is bent, the phase difference is related to the radius of curvature R, the length l of the fiber bend C and the incident wavelength λ as shown in Equation (5): S24. Establish a matrix model of the magnetic field effect: When the polarization-maintaining fiber is affected by a magnetic field, the phase difference is related to magnetic field parameters such as the magnetic field quantity H along the fiber axis component and the magnetic field action length l M as shown in Equation (6): S25. Establish a coupling effect matrix model: When the polarization-maintaining fiber is affected by temperature, magnetic field, and fiber bending, the change in the phase difference is shown in Equation (7):

3. The matrix method for polarization transmission analysis of polarization-maintaining optical fiber according to claim 2, characterized in that The specific content of S22 is as follows: S221. Under the action of temperature on the polarization-maintaining fiber, the birefringence change in the polarization-maintaining fiber is shown in Equation (8): Among them, Δα is the difference between the thermal expansion coefficients of the stress area and the core of the panda-type polarization-maintaining fiber, E is the Young's modulus, C is the photoelastic coefficient, ε is the ellipticity of the core of the polarization-maintaining fiber ε = a / b, where a and b are the radii of the slow and fast axes of the core respectively, υ is the Poisson's ratio coefficient of the panda-type polarization-maintaining fiber, ΔT is the temperature change, n x is the refractive index of the slow axis (X-axis), n y is the refractive index of the fast axis (Y-axis); S222, when the temperature effect length is l T , and the incident light wavelength is λ, the polarization-maintaining fiber in the temperature effect will cause the phase difference between the two modes to be Equation (9): S223. Substitute Equation (8) into Equation (9) to obtain the influence of temperature on the phase difference as shown in Equation (10): S224. Substitute Equation (10) into Equation (2) to obtain the relationship between the phase difference, the temperature change ΔT, the temperature effect length l T and the incident light wavelength λ, and further obtain the matrix model of the temperature effect as Equation (4):

4. The matrix method for polarization transmission analysis of polarization-maintaining optical fiber according to claim 2, characterized in that The specific content of S23 is as follows: S231. When the polarization-maintaining fiber is bent, the birefringence difference is shown in Equation (11): Where: n is the refractive index of the core material, υ is the Poisson's ratio coefficient, p2, p1 are the photoelastic tensors, A is the outer diameter of the polarization-maintaining fiber, and R is the magnitude of the radius of curvature; S232, when the length of the bent part of the optical fiber is l C , when the wavelength of the incident light is λ, the phase difference between the two modes caused by the bending effect of the optical fiber is given by Equation (12): Substitute Equation (12) into Equation (11) to obtain the influence of fiber bending on the phase difference as shown in Equation (13): Substitute Equation (13) into Equation (2), and the relationship between the phase difference and the curvature radius R, the fiber bending length l C and the incident wavelength λ is given by Equation (5):

5. The matrix method for polarization transmission analysis of polarization-maintaining optical fiber according to claim 2, characterized in that The specific content of S24 is as follows: S241. When the polarization-maintaining fiber is affected by a magnetic field, the birefringence difference of the polarization-maintaining fiber is shown in Equation (14): Where λ is the wavelength of the incident light, V is the Verdet constant, and H is the magnetic field quantity of the fiber axial component; S242. When the magnetic field length is l M The phase difference between the two modes of the polarization-maintaining fiber under the action of the magnetic field is given by Equation (15): S243. Substitute Equation (14) into Equation (15) to obtain the influence of the magnetic field on the phase difference as shown in Equation (16): Δδ M = 2VHl M (16); S244. Substitute Equation (16) into Equation (2), and the relationship between the phase difference and the magnetic field parameters such as the magnetic field quantity H of the axial component along the optical fiber and the magnetic field action length l is Equation (6): M ​ 6. The matrix method for polarization transmission analysis of polarization-maintaining optical fiber according to claim 2, wherein The specific content of S25 is as follows: S251. The length L of the polarization-maintaining fiber is decomposed into countless Δl, and the common influence factor is regarded as the individual action under each small segment Δl. The relationship between the components is shown in Equation (17): J = J n …J3J2J1(17); S252. Expand each part of the matrix component to obtain Equation (18): S253. Conduct mathematical calculations on Equation (18) to obtain Equation (19): S254. By combining the influencing factors of temperature, magnetic field, and fiber bending, obtain Equation (20): S255. When the polarization-maintaining optical fiber is affected by temperature, magnetic field, and fiber bending, the change in the phase difference is given by Equation (7):

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