Mode conversion device and design method

Abstract

The present invention aims to provide a mode conversion device and a design method thereof, which can design arbitrary coupling efficiency and full width at half maximum. The present invention is a mode conversion device having a long-period grating in a core of an optical fiber capable of propagating light in at least two propagation modes, characterized in that the long-period grating satisfies a relationship of mathematical formula C1. Wherein, the full width at half maximum FWHM is a wavelength band in which the coupling efficiency is half compared to the coupling efficiency of the mode conversion of the center wavelength, C is the coupling efficiency, L c is the full coupling length, L g is the grating length, Λ is the grating pitch, and Δβ is the difference in propagation constant of the two propagation modes of the center wavelength as a target of mode conversion; [Mathematical formula C1]

Description

Technical Field

[0001] This disclosure relates to a mode conversion device and a design method thereof having a long period grating (LPG) formed in an optical fiber. Background Technology

[0002] Non-patent document 1 discloses an LPG that selectively couples only a certain wavelength of light into the cladding mode of an optical fiber.

[0003] Existing technical documents

[0004] Non-patent literature

[0005] Non-patent literature 1: Craig D. Poole et al., “Helical-Grating Two-Mode Fiber Spatial-Mode Coupler”, JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL 9, NO 5, May 1991. Summary of the Invention

[0006] The problem the invention aims to solve

[0007] Although the relationship between coupling efficiency and full width at half maximum (FWHM) is uniquely determined in an LPG, this relationship changes when any of the following values ​​are altered: the full coupling length, fiber structure, or center wavelength. Therefore, mode converters with LPGs face the challenge of arbitrarily designing both coupling efficiency and FWHM. Furthermore, "full width at half maximum" refers to the band where the coupling efficiency for mode conversion is half that of the center wavelength; it is the band in which the mode converter can perform mode conversion.

[0008] To address the aforementioned problems, the present invention aims to provide a mode conversion device and its design method, which can design arbitrary coupling efficiency and full width at half maximum (FWHM).

[0009] The means used to solve the problem

[0010] To achieve the above objectives, the mode conversion device of the present invention is configured to adjust parameters based on the relationship between the wavelength differential of the propagation constant difference Δβ between the two converted modes and the coupling efficiency and the full width at half maximum (FWHM).

[0011] Specifically, the mode conversion device of the present invention is a mode conversion device having a long-period grating in the core of an optical fiber capable of propagating light in at least two propagation modes, characterized in that the long-period grating satisfies the mathematical formula C1.

[0012] [Mathematical expression C1]

[0013]

[0014] Wherein, full width at half maximum (FWHM) is the band with a coupling efficiency of half that of the center wavelength during mode conversion, C is the coupling efficiency, and L is the coupling efficiency. c It is the fully coupled length, L g Λ is the grating length, Λ is the grating spacing, and Δβ is the difference in propagation constants between the two propagation modes of the center wavelength of the mode conversion object.

[0015] By forming a grating with a spacing of Λ and a grating length of L in the core of an optical fiber with a core radius of a and a relative refractive index difference of Δ, a grating is created that satisfies the mathematical formula C1. g The LPG is a mode conversion device that can achieve the desired full width at half maximum (FWHM) and coupling efficiency C at the desired wavelength.

[0016] The specific design method is a method for determining the design parameters of a long-period grating to be set in the core of an optical fiber capable of propagating light in at least two propagation modes, characterized by:

[0017] The fiber core radius a (μm), relative refractive index difference Δ (%), center wavelength λ0 (nm) of the light undergoing mode conversion, coupling efficiency C, and full width at half maximum (FWHM) (nm) of the optical fiber are given.

[0018] Based on the fiber core radius a (μm) and relative refractive index difference Δ (%), the propagation constant difference Δβ and its wavelength derivative dΔβ / dλ of the two propagation modes of the center wavelength λ0 (nm) of the mode conversion object are obtained through mode analysis.

[0019] The grating spacing Λ of the long-period grating is calculated using mathematical formula C2;

[0020] The coefficient b is calculated using mathematical formula C3;

[0021] The full coupling length L is calculated using mathematical formula C4. c ;as well as,

[0022] The grating length L is calculated using mathematical formula C5. g .

[0023] [Mathematical expression C2]

[0024]

[0025] [Mathematical expression C3]

[0026] b = -1169.3C + 1705.6 (C3)

[0027] [Mathematical expression C4]

[0028]

[0029] Among them, the full width at half maximum (FWHM) is the band with a coupling efficiency of half that of the mode conversion coupling amount with the center wavelength λ0.

[0030] [Mathematical expression C5]

[0031]

[0032] Therefore, the present invention can provide a mode conversion device and its design method, which can design arbitrary coupling amount and full width at half maximum (FWHM).

[0033] The mode conversion device of the present invention is characterized in that it further comprises a tapped waveguide, which is located at the rear section of the long-period grating in the direction of light propagation, and outputs light of the desired wavelength propagating in the core of the optical fiber from the side of the optical fiber, which is converted from one mode to another by the long-period grating.

[0034] This mode conversion device can extract light of a desired wavelength at the desired power from light propagating in an optical fiber.

[0035] Here, the mode conversion device of the present invention may be: the long-period grating and the tapped waveguide are arranged longitudinally in the optical fiber.

[0036] In this case, the long-period grating is configured such that the design parameters described in mathematical formula C1 are different for each mode, so that the wavelengths of the light converted from one mode to the other modes are different. This allows light of different wavelengths to be extracted from each tap waveguide.

[0037] In this case, the long-period grating can also be set as follows: the grating length L g The coupling efficiencies are different so that the wavelengths of light converted from one mode to the other modes are the same, but the coupling efficiencies are different. Light with different desired power can be extracted from each tapped waveguide.

[0038] Furthermore, the inventions described above can be combined in combination as much as possible.

[0039] The effects of the invention

[0040] This invention provides a mode conversion device and its design method, which can design arbitrary coupling efficiency and full width at half maximum (FWHM). Attached Figure Description

[0041] Figure 1 This is a diagram illustrating the general structure of a mode conversion device with a long period grating (LPG).

[0042] Figure 2This is a diagram illustrating an example of the transmission spectrum when a long-period grating (LPG) is formed.

[0043] Figure 3 This is a figure illustrating an example of the relationship between coupling and full width at half maximum (FWHM) when a long period grating (LPG) is formed.

[0044] Figure 4 This is a diagram illustrating an example of the relationship between FWHM and coupling quantity.

[0045] Figure 5 This explains FWHM and L c A graph showing the relationship between the product of and dΔβ / dλ.

[0046] Figure 6 This is a graph illustrating the relationship between coupling efficiency C and coefficient b.

[0047] Figure 7 This is a diagram illustrating the design method of the present invention.

[0048] Figure 8 This is a diagram illustrating the mode conversion device of the present invention.

[0049] Figure 9 This is a diagram illustrating the mode conversion device of the present invention.

[0050] Figure 10 This is a diagram illustrating the mode conversion device of the present invention. Detailed Implementation

[0051] Embodiments of the present invention will be described with reference to the accompanying drawings. The embodiments described below are examples of the present invention, and the present invention is not limited to these embodiments. Furthermore, in this specification and the accompanying drawings, elements with the same reference numerals denote elements that are identical to each other.

[0052] (Implementation Method 1)

[0053] Figure 1 This is a diagram illustrating the general structure of a mode conversion device with a long-period grating (LPG). By periodically applying a perturbation, the incident mode (mode 1) is converted to another mode (mode 2). At this time, the difference (Δβ) in the propagation constants of mode 1 and mode 2 with the applied perturbation period (Λ) and center wavelength (λ0) needs to satisfy the relationship of equation (1).

[0054] [Mathematical Expression 1]

[0055]

[0056] Here, the length by which the incident mode 1 is fully coupled to mode 2 is called the full coupling length (Lc). Lc is determined by the mode transition amount at each perturbation point; when the mode transition amount at each point is large, the full coupling length becomes shorter. On the other hand, when the mode transition amount at each point is large, the loss caused by mode mismatch will increase. Furthermore, the grating length is defined as Lg. When Lg = Lc, as described above, ... Figure 1 As shown in (a), mode 1 is completely switched to mode 2. Furthermore, by adjusting Lg, it is possible to achieve the following: Figure 1 (b) shows the amount of coupling from mode 1 to mode 2 (C).

[0057] The coupling efficiency C of wavelength λ0 at this time is expressed by formula (2).

[0058] [Mathematical Expression 2]

[0059]

[0060] When Lg = Lc, mode 1 is fully coupled to mode 2, and C = 1.

[0061] Figure 2 This figure illustrates an example of the transmittance of light with a center wavelength (λ0) of 1350 nm in an optical fiber with a core radius (a) of 4.5 μm and a relative refractive index difference (Δ) between the core and cladding of 0.35%, when a grating with an Lc = 3 cm is formed. The figures show examples of transmittance. Figure 1 Modes 1 and 2 described are LP01 mode and LP11a mode, respectively. Here, "transmittance" refers to the transmittance of the LP01 mode when incident. "Coupling efficiency" is the amount of attenuation at the center wavelength. Figure 2 The lines represent the transmittance when Lg / Lc is changed. It can be seen that the coupling efficiency can be controlled by controlling Lg / Lc, and the bandwidth tends to narrow when the coupling efficiency increases (making Lg larger).

[0062] Figure 3 This is a graph illustrating the relationship between coupling efficiency and full width at half maximum (FWHM). Let λ0 = 1350 nm and Lc = 3 cm. Let the core radius a and the relative refractive index Δ of the fiber structure be (a = 5 μm, Δ = 0.28%), (a = 4.5 μm, Δ = 0.35%), and (a = 4 μm, Δ = 0.44%), respectively.

[0063] Depend on Figure 3It is known that as the coupling efficiency increases (by increasing Lg), the FWHM decreases, and the relationship between FWHM and coupling efficiency depends on the fiber structure. The relationship between FWHM and coupling efficiency changes not only with the fiber structure but also with changes in Lc or λ0. Here, the wavelength derivative (dΔβ / dλ) of the two-mode propagation constant difference Δβ at wavelength λ0 is introduced as a parameter representing the fiber structure. Furthermore, the two-mode propagation constant difference Δβ can be determined through mode analysis based on the fiber structure.

[0064] Figure 4 With dΔβ / dλ as the horizontal axis, Figure 3 The graph illustrating the relationship between FWHM and coupling efficiency has been redrawn. In this graph, data for different fiber structures (Δ, a) and center wavelength (λ0) are shown overlapping. Figure 4 It can be seen that the relationship between FWHM and dΔβ / dλ is independent of the fiber structure and center wavelength, and is uniquely determined for the coupling efficiency.

[0065] Furthermore, by calculating the product of FWHM and Lc, the relationship between FWHM and dΔβ / dλ is independent of Lc, and is uniquely determined for the coupling efficiency. Figure 5 This is a graph illustrating the relationship between the product of FWHM and Lc and dΔβ / dλ. Each curve represents the calculated relationship between the product of FWHM and Lc and dΔβ / dλ, and the dashed line represents the approximate function obtained for each coupling efficiency. The approximate functions for each coupling efficiency are shown in equations (3) to (6).

[0066] [Mathematical Expression 3]

[0067]

[0068] [Mathematical Expression 4]

[0069]

[0070] [Mathematical Expression 5]

[0071]

[0072] [Mathematical Expression 6]

[0073]

[0074] As shown in equations (3) to (6), it can be seen that the relationship between the product of FWHM and Lc and dΔβ / dλ can be approximated by the inverse proportional equation shown in equation (7).

[0075] [Mathematical Expression 7]

[0076]

[0077] Here, the coupling efficiency is related to the inversely proportional coefficient b as follows: Figure 6 As shown. According to Figure 6 The coefficient b is obtained according to equation (8).

[0078] [Mathematical Expression 8]

[0079] b = -1169.3C + 1705.6 (8)

[0080] Here, C is the coupling efficiency in linear representation.

[0081] Therefore, by setting a grating structure in the optical fiber that satisfies equations (1), (2), (7), and (8), a mode conversion device with arbitrary coupling efficiency and bandwidth can be constructed. That is, the mode conversion device of the present invention has a long-period grating in the core of an optical fiber capable of propagating light in at least two propagation modes, characterized in that the long-period grating satisfies the mathematical relationship C1.

[0082] [Mathematical expression C1]

[0083]

[0084] Wherein, full width at half maximum (FWHM) is the band with a coupling efficiency of half that of the center wavelength during mode conversion, C is the coupling efficiency, and L is the coupling efficiency. c It is the fully coupled length, L g Λ is the grating length, Λ is the grating spacing, and Δβ is the difference in propagation constants between the two propagation modes of the center wavelength of the mode conversion object.

[0085] Although the conversion efficiency of LP01 and LP11 modes is used as an example for calculation here, the same approach applies to coupling between other modes, such as LP01 and cladding modes, or LP01 and LP02 modes, LP11 and LP21 modes, etc. Furthermore, regardless of whether the fiber is a step-index structure, the same approach can be used for other structures such as graded-index structures.

[0086] (Implementation Method Two)

[0087] Figure 7 This is a flowchart illustrating a method for designing a long-period grating for a mode conversion device as described in Embodiment 1. This design method determines the design parameters of a long-period grating to be disposed in the core of an optical fiber capable of propagating light in at least two propagation modes, characterized by:

[0088] The fiber core radius a (μm), relative refractive index difference Δ (%), center wavelength λ0 (nm) of the light undergoing mode conversion, coupling efficiency C, and full width at half maximum (FWHM) (nm) of the optical fiber are given (step S01).

[0089] Based on the fiber core radius a (μm) and relative refractive index difference Δ (%), the propagation constant difference Δβ and its wavelength differential dΔβ / dλ of the two propagation modes of the center wavelength λ0 (nm) of the mode conversion object are obtained by mode analysis (step S02).

[0090] The grating spacing Λ of the long-period grating is calculated using mathematical formula C2 (step S03);

[0091] Calculate the coefficient b using mathematical formula C3 (step S04);

[0092] The full coupling length L is calculated using mathematical formula C4. c (Step S05); and,

[0093] The grating length L is calculated using mathematical formula C5. g (Step S06).

[0094] [Mathematical expression C2]

[0095]

[0096] [Mathematical expression C3]

[0097] b = -1169.3C + 1705.6 (C3)

[0098] [Mathematical expression C4]

[0099]

[0100] Among them, full width at half maximum (FWHM) is a band with a coupling efficiency of half that of the mode conversion at the center wavelength λ0.

[0101] [Mathematical expression C5]

[0102]

[0103] The design parameters represent the fiber structure's dΔβ / dλ, Lc, Lg, Λ, λ0, C, and FWHM. First, in step S01, the fiber core radius a (μm), relative refractive index difference Δ (%), center wavelength λ0 (nm) of the light undergoing mode conversion, coupling efficiency C, and full width at half maximum (FWHM) (nm) are given as specification values.

[0104] In step S02, based on the fiber core radius a (μm) and relative refractive index difference Δ (%) of the fiber structure, the propagation constant difference Δβ and its wavelength derivative dΔβ / dλ of the center wavelength λ0 (nm) are obtained through mode analysis.

[0105] In step S03, the propagation constant difference Δβ is substituted into equation (C2) to calculate the grating spacing Λ (μm).

[0106] In step S04, the coupling efficiency C, which is the specification value, is substituted into equation (C3) to calculate the coefficient b.

[0107] In step S05, the coefficient b, the full width at half maximum (FWHM) of the specification value, and the wavelength differential dΔβ / dλ are substituted into equation (C4) to calculate the full coupling length Lc.

[0108] In step S06, the grating length Lg is calculated by substituting the fully coupled length Lc and the coupling efficiency C (as a specification value) into equation C5.

[0109] In addition, steps S03 and steps (S04 to S06) can be performed simultaneously, or one of them can be performed first.

[0110] The mode-convertible wavelength range (FWHM) varies depending on the device using the LPG. For example, a wider range is preferred in cases of wideband applications such as mode-multiplexed transmission. On the other hand, a narrower range is preferred in the case of tapped devices as described in Embodiment 3. The key point of the design method in this embodiment is that it is possible to derive the grating pitch Λ and grating length Lg that conform to the intended use (specifications) of the device.

[0111] (Implementation Method 3)

[0112] Figure 8 This is a diagram illustrating the mode conversion device 301 of this embodiment. The mode conversion device 301 is characterized by further comprising a tapped waveguide 53, which is located at the rear end of the long-period grating 21 in the direction of light propagation, and outputs light of the desired wavelength from the core 51 of the optical fiber 50, which is the light from the side of the optical fiber 50 in which the long-period grating 21 is converted from one mode to other modes.

[0113] In this embodiment, one mode is designated as the basic mode, and the other modes are designated as higher-order modes for explanation.

[0114] The mode conversion device 301 includes: a tap 10, which forms a tap waveguide 53, and the tap waveguide 53 outputs the higher-order mode of light propagating in the core 51 of the optical fiber 50 from the side of the optical fiber 50.

[0115] The grating section 20 is located at the front end of the tap head 10 in the direction of light propagation, and a grating 21 is formed on the core 51 of the optical fiber 50 to convert light of the desired wavelength from the basic mode to the higher-order mode.

[0116] The optical fiber 50 is a step-index optical fiber. A grating section 20 and a tapped section 10 are sequentially formed along the length of the optical fiber 50. The direction in which light can incident on the tapped waveguide 53 is defined as the waveguide direction. Figure 8 In this configuration, the optical waveguide direction is from left to right. Furthermore, the tap direction of the tapped waveguide 53, from the fiber core 51 towards the side of the optical fiber 50, is defined as the tap direction. Figure 8 In this context, the tap direction is the direction that is tilted in the positive direction relative to the optical waveguide direction.

[0117] The grating section 20 uses a long-period grating 21 to convert the desired wavelength of light propagating in the core 51 of the optical fiber 50 from the LP01 mode to the LP11 mode by only the desired amount. The grating structure can be realized, for example, by femtosecond laser processing, CO2 laser processing, or pressing of the grating.

[0118] The tap 10 has a tapped waveguide 53 extending from the center of the fiber core 51 at an angle α toward the side of the fiber 50 (the interface of the cladding 52). The tap 10 is controlled by adjusting the angle α between the tapped waveguide 53 and the fiber core 51, and the diameter d of the tapped waveguide 53. t And the refractive index of the tapped waveguide 53, selectively extracting only the LP11 mode from the core 51.

[0119] Here, the light coupled from fiber core 51 to tapped waveguide 53 is defined as tapped light, and the light that propagates unchanged in fiber core 51 is defined as transmitted light. For example, by connecting receiver 54 to the output end (side of fiber 50) of tapped head 10, it is possible to extract and receive only tapped light from fiber 50.

[0120] In the tap 10, the coupling efficiency from the fiber core 51 to the tap waveguide 53 is highly dependent on the propagation mode of the light propagating in the fiber core 51. This is because higher-order modes are less restricted and more easily coupled to the tap waveguide 53. Therefore, it is possible to transfer only higher-order modes to the tap waveguide.

[0121] Here, in order to couple only the higher-order modes with the tapped waveguide 53, the refractive index and diameter d of the tapped waveguide 53 are... t The values ​​of are important. If these values ​​are too large, the NA of the tapped waveguide 53 increases, making it easier for the LP01 mode to couple, thus increasing the loss of transmitted light. On the other hand, if these values ​​are too small, the NA of the tapped waveguide 53 decreases, making it difficult for higher-order modes to couple, thus reducing the coupling efficiency of the tapped light to the tapped waveguide 53. In other words, the refractive index and diameter d of the tapped waveguide 53 need to be appropriately determined. t The value of .

[0122] Furthermore, to ensure efficient coupling of higher-order modes with the tapped waveguide 53 and to allow the fundamental mode light to propagate intact within the fiber core 51, α needs to be sufficiently small, and the mode migration needs to be adiabatic. If α is large, the LP01 mode will also be affected by the tapped waveguide 53 and coupled to the radiation mode, resulting in loss. Therefore, the upper limit of α is determined from the perspective of the loss of the LP01 mode. On the other hand, although α can be any value greater than 0, since α determines the total length L of the tap 10... tap Therefore, the lower limit of α is determined based on the propagation loss of the tapped waveguide 53 and the requirements for the overall length of the equipment.

[0123] In a typical single-mode optical fiber, the diameter d of fiber 50 f For example, to make the suction head L 125 μm tap For values ​​below 5cm, α needs to be set to 0.07° or higher.

[0124] The grating section 20 has a grating 21 with a spacing of Λ. For example, the grating 21 is a long-period fiber grating (LPG). In order to convert light of arbitrary wavelength λ and desired light amount from LP01 mode to LP11 mode in the grating section 20, the grating 21 is shaped using the design parameters described in Embodiments 1 and 2.

[0125] (Implementation Method 4)

[0126] Figure 9 as well as Figure 10 This is a diagram illustrating the mode conversion device 302 of this embodiment. The mode conversion device 302 is characterized in that it converts... Figure 8 The groups of long-period gratings 21 and tapped waveguides 53 in the described mode conversion device 301 are arranged longitudinally in the optical fiber 50. That is, the mode conversion device 302 is a system composed of multiple segments of the mode conversion device 301. The groups of long-period gratings 21 and tapped waveguides 53 (mode conversion device 301) are configured, for example, at certain intervals (a few meters to a few kilometers).

[0127] Figure 9 The long-period grating 21 of the mode conversion device 302 is characterized in that the design parameters recorded in the mathematical formula C1 are different, so that the wavelengths of the light converted from the one mode to the other mode are different from each other.

[0128] Figure 9 The mode conversion device 302 assigns wavelengths to each mode conversion device (301-1, 301-2, 301-3) and controls the extracted signal according to the wavelength. By setting the design parameters of each mode conversion device according to the wavelength interval (center wavelength λ0 and bandwidth (FWHM)) of the wavelength to be extracted, signals can be extracted at arbitrary wavelength intervals.

[0129] Figure 10 The long-period grating 21 of the mode conversion device 302 is characterized in that the grating length Lg is different for each mode, so that the wavelength of the light converted from the one mode to the other mode is the same while the coupling efficiency is different.

[0130] Figure 10 The mode conversion device 302 assigns multiple mode conversion devices (301-1, 301-2, 301-3) to a single wavelength. By transmitting signals at a single wavelength and controlling the coupling efficiency of each LPG, signals can be extracted bit by bit in multiple segments. Figure 10 The mode conversion device 302 can output signals at multiple locations simultaneously with a single wavelength.

[0131] Explanation of reference numerals in the attached figures

[0132] 10: Head extraction

[0133] 20: Grating section

[0134] 21: Long Period Grating (LPG)

[0135] 50: Fiber optic cable

[0136] 51: Fiber Core

[0137] 52: Cladding

[0138] 53: Tapped waveguide

[0139] 54: Light receiver

[0140] 301, 302: Mode conversion equipment.

Claims

1. A mode conversion device, comprising a long-period grating in the core of a step-index optical fiber, for converting the mode of light propagating in the optical fiber from one mode to other modes, characterized in that: The long-period grating satisfies the mathematical formula C1; [Mathematical expression C1] Wherein, full width at half maximum (FWHM) is the band with a coupling efficiency of half that of the center wavelength during mode conversion, C is the coupling efficiency, and L is the coupling efficiency. c It is the fully coupled length, L g Λ is the grating length, Δβ is the grating spacing, Δβ is the difference in propagation constants between the two modes at the center wavelength of the mode conversion object, and λ is the wavelength.

2. The mode conversion device according to claim 1, characterized in that, It also has a tapped waveguide located at the rear end of the long-period grating in the direction of light propagation, which outputs the desired wavelength of light propagating in the optical fiber from the side of the optical fiber, which is the wavelength from which the long-period grating changes from one mode to another.

3. The mode conversion device according to claim 2, characterized in that, The long-period grating and the tapped waveguide are arranged longitudinally in the optical fiber.

4. The mode conversion device according to claim 3, characterized in that, In the long-period grating, for the design parameters recorded in mathematical formula C1, the same parameter in each group of design parameters is different, so that the wavelengths of the light converted from the 1 mode to the other modes are different from each other.

5. The mode conversion device according to claim 3, characterized in that, In the long-period grating, the grating length L g They are different, so that the wavelength of the light converted from the one mode to the other modes is the same, but the coupling efficiency is different.

6. A design method for determining the design parameters of a long-period grating, wherein the long-period grating is installed in the core of the optical fiber to convert the mode of light propagating in the optical fiber from one mode to other modes, characterized in that: The fiber core radius *a*, the relative refractive index difference *Δ* between the fiber core and the fiber cladding (in %), the center wavelength *λ0* of the light undergoing mode conversion, the coupling efficiency *C*, and the full width at half maximum (FWHM) (in nm) are given. The unit of core radius *a* is *μm*, the unit of relative refractive index difference *Δ* is %, the unit of center wavelength *λ0* is *nm*, and the unit of FWHM is *nm*. Based on the fiber core radius *a* and relative refractive index difference *Δ*, mode analysis is used to obtain the propagation constant difference *Δβ* and its wavelength derivative *dΔβ / dλ* between the two modes at the center wavelength *λ0* of the mode conversion target. The grating spacing Λ of the long-period grating is calculated using mathematical formula C2; The coefficient b is calculated using mathematical formula C3; The full coupling length L is calculated using mathematical formula C4. c ;as well as, The grating length L is calculated using mathematical formula C5. g ; [Mathematical expression C2] [Mathematical expression C3] [Mathematical expression C4] Among them, the full width at half maximum (FWHM) is a band with a coupling efficiency of half that of the mode conversion at the center wavelength λ0. [Mathematical expression C5] 。

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