Wavelength-Tunable, Ultra-Broadband, Biconical, Long-Period Fiber Grating Mode Converter Based on the Dual-Resonance Effect

We demonstrated a wavelength-tunable, ultra-wideband, biconical, long-period fiber grating (BLPFG) mode converter in a two-mode fiber based on fusion taper technology and CO2 laser writing technology. Theoretical and experimental results show that after changing the diameter of the two-mode fiber by fusing and tapering, the dispersion turning point of the fiber is adjusted and wavelength-tunable broadband mode conversion is achieved efficiently. Theoretical simulation shows that the mode conversion bandwidth can cover the O + E + S + C band. In the experiment, we fabricated adiabatic tapers with cladding diameters of 113 μm and 121 μm and wrote gratings on these tapers to achieve dual-resonance coupling, thus realizing mode conversion from LP01 to LP11, with a 15 dB bandwidth of 148.8 nm from 1229.0 nm to 1377.8 nm and of 168.5 nm from 1319.7 nm to 1488.2 nm, respectively. As far as we know, this is the first time that fusion taper technology has been used to adjust the window of the dual-resonant coupling of an optical fiber. This work broadens the scope of application of the dual-resonance effect and proposes a general method for widening the bandwidth of a fiber grating with tunable wavelength.

For decades, fiber gratings have received widespread attention due to their ease of preparation and high efficiency. So far, linearly polarized (LP) mode conversion from LP 01 to LP 11 , LP 21 , LP 31 , and LP 02 using a fiber grating has been realized [23][24][25]. However, the fiber grating suffered from its narrow bandwidth due to the inherent influence of the working mechanism. A lot of work has been devoted to expanding the working bandwidth of fiber gratings [5,10,11,17,26]. Rottwitt used chirped gratings to increase the 20 dB mode conversion bandwidth by 4.8 times (8.6 nm) [27]. Zhao et al. cascaded three linear, length-apodized, phase-shifted, long-period gratings to achieve a 10 dB bandwidth of 182.0 nm [28]. Guo et al. used the double-resonance effect to obtain a 15 dB bandwidth of 118.2 nm [29]. The performance comparison of these mode converters is shown in Table 1. Compared with other methods, the dual-resonance effect does not need to design a complicated grating structure, nor does it require a special processing technology. It is the simplest method with the best performance. However, the dual-resonance effect can only broaden the bandwidth of the grating at a specific wavelength range determined by the fiber parameters, which limits the application of the dual-resonance effect. In this paper, we demonstrated the production of a wavelength-tunable, ultra-wideband, biconical, long-period fiber grating (BLPFG) mode converter in a two-mode fiber based on fusion taper technology and CO 2 laser writing technology. Theoretical and experimental results show that there is dual-resonance coupling at the dispersion turning point of the two-mode fiber, which effectively expands the working bandwidth of the grating. By changing the diameter of the fiber by fusing and tapering, the dispersion turning point of the fiber is adjusted and broadband mode conversion is achieved efficiently. Theoretical simulation shows that the mode conversion bandwidth can cover the O + E + S + C (O: 1270~1370 nm, E: 1370~1460 nm, S: 1460~1530 nm, C: 1530~1565 nm) band. In the experiment, we made adiabatic tapers with cladding diameters of 113 µm and 121 µm and wrote gratings on the taper to achieve dual-resonance coupling. The fabricated BLPFGs realized the mode conversion from LP 01 to LP 11 with a 15 dB of 148.8 nm (1229.0~1377.8 nm) and 168.5 nm (1319.7~1488.2 nm), respectively. The size of the bandwidth can be adjusted by changing the grating period so as to realize the large-bandwidth mode conversion with adjustable wavelength. This work broadens the scope of application of the dual-resonance effect and proposes a general method for widening the bandwidth of a fiber grating with tunable wavelength.

Theory and Simulation Results
The fiber used in this work was a step-index two-mode fiber (TMF) (SM2000, Thorlabs) with a core diameter of 11 µm and a cladding diameter of 125 µm. The numerical aperture (NA) of the fiber is 0.12. We used the finite element method (FEM) to simulate the fiber mode dispersion curves, and the results are shown in Figure 1. The results show that the fiber supports LP 01 and LP 11 modes from 1000.0 nm to 1750.0 nm. The effective index difference of the two modes was above 10 −4 in the simulation wavelength range. Thus, the fiber was used as a robust TMF in this work. Coupled mode theory [30] demonstrates that the grating period of an LPFG is inversely proportional to the effective index difference between the fundamental mode and the target mode: where Λ is the period of the LPFG, λ is the resonant wavelength of the LPFG, and , and , are the effective refractive index of LP01 and LP11 modes, respectively.
The coupling coefficient of LP01 and LP11 modes can be expressed as a function of the refractive index modulation Δ , where is the circular frequency of the light wave and and are the normalized mode fields of LP01 and LP11, respectively. Then the transmission spectrum of the LP01 mode of an LPFG is [29] 1 4 sin 4 where the coupling length / 2| | and the parameter of phase mismatch , 1,2,3, … In general, we set 1.
We calculated the period of the TMF-LPFG versus different wavelengths using Equation (1), and the results present a non-monotonic variation trend of parabola, as illustrated in Figure 2a. The LPFG period first fell and then rose as the wavelength increased, and reached the minimum of 559.4 μm at a wavelength of 1444.0 nm, which is called the dispersion turning point. That is, one grating period value corresponds to two resonant wavelengths when the grating period is larger than the value at the dispersion turning point. While the grating period is near the dispersion turning point, the two resonant wavelengths get close and merge with each other, and then the bandwidth that satisfies the mode conversion condition doubles [29,31]. Figure 2b shows the simulated spectra of the LPFGs with different periods marked in Figure 2a according to Equation (3). The spectrum lines and their corresponding period markers share the same color. When the period was 570.0 μm as the purple lines show, the two resonance wavelengths were 1257.0 nm and 1620.0 nm and the resonance dips appeared at the two wavelengths in the spectrum, and their 15 dB bandwidths were 29.0 nm and 26.0 nm, respectively. When the period was reduced, as the orange and yellow Coupled mode theory [30] demonstrates that the grating period of an LPFG is inversely proportional to the effective index difference between the fundamental mode and the target mode: λ n e f f ,01 − n e f f ,11 (1) where Λ is the period of the LPFG, λ is the resonant wavelength of the LPFG, and n e f f ,01 and n e f f ,11 are the effective refractive index of LP 01 and LP 11 modes, respectively. The coupling coefficient of LP 01 and LP 11 modes can be expressed as a function of the refractive index modulation ∆ε q , where ω is the circular frequency of the light wave and → E 01 and → E 11 are the normalized mode fields of LP 01 and LP 11 , respectively . Then the transmission spectrum of the LP 01 mode of an LPFG is [29] P 01 (z) = 1 − π 2 π 2 + 4L 2 c δ 2 sin 2 L c where the coupling length L c = π/(2|κ|) and the parameter of phase mismatch δ = . . In general, we set q = 1. We calculated the period of the TMF-LPFG versus different wavelengths using Equation (1), and the results present a non-monotonic variation trend of parabola, as illustrated in Figure 2a. The LPFG period first fell and then rose as the wavelength increased, and reached the minimum of 559.4 µm at a wavelength of 1444.0 nm, which is called the dispersion turning point. That is, one grating period value corresponds to two resonant wavelengths when the grating period is larger than the value at the dispersion turning point. While the grating period is near the dispersion turning point, the two resonant wavelengths get close and merge with each other, and then the bandwidth that satisfies the mode conversion condition doubles [29,31].
Sensors 2021, 21, 5970 4 of 10 and the 15 dB bandwidth reached the maximum value of 204.0 nm when the period was 561.0 μm. When the period was further reduced to 559.4 μm (blue line), the two dips merged into a broadband single dip with a 15 dB bandwidth of 144.0 nm. Therefore, to achieve the largest-possible bandwidth, a period slightly larger than the corresponding dispersion turning point should be selected. The detailed situation mentioned above is shown in Table 2.   Now we can only achieve broadband mode conversion around 1444.0 nm. The reason is that the dispersion turning point of the optical fiber depends on the inherent structure of the optical fiber. For any kind of few-mode fiber, the dispersion turning point corresponding to the desired mode conversion is fixed, that is, we cannot take advantage of the dual-resonance effect at a wavelength away from the dispersion turning point.
The key is to change the optical structure of the fiber. Equation (1) shows that the grating period Λ is a function of the mode effective index , and , . If we change the geometric parameters of the fiber, the value of , , also changes. Thus, the curve of the grating period Λ in the wavelength domain deforms and the turning point changes its position. In other words, the dispersion turning point moves in the wavelength domain by changing the structure of the optical fiber so that the dual-resonance effect can be used in other bands. As the simplest method, fusing and tapering technology is chosen to change the diameter of the core and cladding. When the fiber is stretched to a certain ratio, the characteristics of gratings with different periods can be simulated by the same method.
Here we focused on the influence of different stretching ratios on the dispersion turning points and grating bandwidths. As shown in Figure 3a, as the diameter of the fiber became smaller, the dispersion turning point moved from the initial value of 1444.0 nm to the shortwave direction. The change in the dispersion turning point along the fiber diameter was almost linear, and the R-square (determination coefficient) of the linear fit was  When the period was further reduced to 559.4 µm (blue line), the two dips merged into a broadband single dip with a 15 dB bandwidth of 144.0 nm. Therefore, to achieve the largest-possible bandwidth, a period slightly larger than the corresponding dispersion turning point should be selected. The detailed situation mentioned above is shown in Table 2. Table 2. Characteristic parameters of simulated spectra corresponding to different periods of the LPFG. Now we can only achieve broadband mode conversion around 1444.0 nm. The reason is that the dispersion turning point of the optical fiber depends on the inherent structure of the optical fiber. For any kind of few-mode fiber, the dispersion turning point corresponding to the desired mode conversion is fixed, that is, we cannot take advantage of the dual-resonance effect at a wavelength away from the dispersion turning point.
The key is to change the optical structure of the fiber. Equation (1) shows that the grating period Λ is a function of the mode effective index n e f f ,01 and n e f f ,11 . If we change the geometric parameters of the fiber, the value of n e f f ,01 − n e f f ,11 also changes. Thus, the curve of the grating period Λ in the wavelength domain deforms and the turning point changes its position. In other words, the dispersion turning point moves in the wavelength domain by changing the structure of the optical fiber so that the dual-resonance effect can be used in other bands. As the simplest method, fusing and tapering technology is chosen to change the diameter of the core and cladding. When the fiber is stretched to a certain ratio, the characteristics of gratings with different periods can be simulated by the same method.
Here we focused on the influence of different stretching ratios on the dispersion turning points and grating bandwidths. As shown in Figure 3a, as the diameter of the fiber became smaller, the dispersion turning point moved from the initial value of 1444.0 nm to the shortwave direction. The change in the dispersion turning point along the fiber diameter was almost linear, and the R-square (determination coefficient) of the linear fit was 1.0000. The movement of the dispersion turning point also caused a change in the bandwidth, as shown in Figure 3b. The bandwidth of the grating decreased when the dispersion turning point blue-shifted. Similarly, the change was linear, and the determination coefficient was 0.9992. The simulation results show that the position of the dispersion turning point can be adjusted by tapering the fiber. For the fiber used in this work, its dispersion turning point moved from the initial value of 1444.0 nm to the shortwave direction and its mode conversion bandwidth could cover the O + E + S + C band by adjusting the diameter of the fiber.  The marked points in Figure 3 corresponding to the cladding diameter of 125 μm, 121 μm, and 113 μm were then taken into the experiment to achieve dual-resonance broadband mode conversion surrounding 1444.0 nm, 1398.0 nm, and 1306.0 nm, respectively. The fiber grating fabrication process included two steps. First, the optical fiber was stretched adiabatically to the desired size. The equipment used for tapering was a Vytran GPX 3850 Glass Processor Workstation. The flame scan width was set to 20 mm, and the stretched length was 2.00 mm and 0.66 mm for the final diameter of 113 μm and 121 μm, respectively. The obtained fiber taper was composed of a waist with a length of L1 and a taper area with a length of L2, as shown in Figure 4b. For the fabricated taper in this work, L1 was 20 mm, which is long enough for the fiber grating fabrication. L2 was approximately equal to the stretching length of the fiber, which is 2.00 mm and 0.66 mm, respectively. Real-time power monitoring during processing shows that the fiber taper is flat enough to ensure a low transmission loss below 0.1 dB.
After obtaining the adiabatic taper, we used a CO2 laser (Han's CO2-H30) to process the long-period grating at the waist of the taper. The processing device is shown in Figure  4a. Both ends of the adiabatic taper were fused to a single-mode fiber (SMF). A supercontinuum light source (SC-5-FC 480-2200 nm) and a spectrum analyzer (YOKOGAWA AQ6370D) were connected to the ends of the SMF to monitor the transmittance of the fundamental mode in the BLPFG. A 15 g counterweight was attached to the optical fiber to ensure constant tension during processing. The taper waist was single-side-exposed by the CO2 laser pulse with a power of 2.1 W. When the optical fiber was irradiated, the glass melted and deformed slightly under the action of tension, forming a refractive index modulation. Usually, the fiber needs to be exposed repeatedly several times to achieve the best performance. Figure 4b shows a schematic diagram of the obtained BLPFG and a photomicrograph of one period in the taper waist. The CO2 laser side irradiation formed a trench with a depth of 10.0 nm and a width of 50.0 nm. The marked points in Figure 3 corresponding to the cladding diameter of 125 µm, 121 µm, and 113 µm were then taken into the experiment to achieve dual-resonance broadband mode conversion surrounding 1444.0 nm, 1398.0 nm, and 1306.0 nm, respectively. The fiber grating fabrication process included two steps. First, the optical fiber was stretched adiabatically to the desired size. The equipment used for tapering was a Vytran GPX 3850 Glass Processor Workstation. The flame scan width was set to 20 mm, and the stretched length was 2.00 mm and 0.66 mm for the final diameter of 113 µm and 121 µm, respectively. The obtained fiber taper was composed of a waist with a length of L1 and a taper area with a length of L2, as shown in Figure 4b. For the fabricated taper in this work, L1 was 20 mm, which is long enough for the fiber grating fabrication. L2 was approximately equal to the stretching length of the fiber, which is 2.00 mm and 0.66 mm, respectively. Real-time power monitoring during processing shows that the fiber taper is flat enough to ensure a low transmission loss below 0.1 dB.

Results and Discussion
We first confirmed that the dual-resonance effect is still effective on the tapered fiber. Figure 5 shows the transmission spectra of the BLPFGs with a cladding diameter of 121 μm and periods of 568.0 μm, 566.0 μm, 564.0 μm, and 562.0 μm. Each spectrum was filtered by FFT to remove the slight interference of LP01 and LP11 caused by the fusion of SMF and TMF. There were two resonance dips when the period was 568.0 μm. As the grating period decreases, the two resonant dips approach each other gradually, and the bridge between the two dips decreases. When the period was reduced to 562.0 μm, the two dips combined into a broadband dip and the 15 dB bandwidth reached the maximum of 168.5 nm from 1319.7 nm to 1488.2 nm. Thus, as the period decreases, the 15 dB bandwidth increases. The experimental spectra show the same change trend as the simulation results in Figure 2b. The origin dispersion turning point of the used fiber was located at 1474.0 nm. After tapering to 121 μm and 113 μm, we moved the dispersion turning point to a shorter wavelength and realized broadband mode conversion. Figure 6 shows the conversion efficiency After obtaining the adiabatic taper, we used a CO 2 laser (Han's CO 2 -H30) to process the long-period grating at the waist of the taper. The processing device is shown in Figure 4a. Both ends of the adiabatic taper were fused to a single-mode fiber (SMF). A supercontinuum light source (SC-5-FC 480-2200 nm) and a spectrum analyzer (YOKOGAWA AQ6370D) were connected to the ends of the SMF to monitor the transmittance of the fundamental mode in the BLPFG. A 15 g counterweight was attached to the optical fiber to ensure constant tension during processing. The taper waist was single-side-exposed by the CO 2 laser pulse with a power of 2.1 W. When the optical fiber was irradiated, the glass melted and deformed slightly under the action of tension, forming a refractive index modulation. Usually, the fiber needs to be exposed repeatedly several times to achieve the best performance. Figure 4b shows a schematic diagram of the obtained BLPFG and a photomicrograph of one period in the taper waist. The CO 2 laser side irradiation formed a trench with a depth of 10.0 nm and a width of 50.0 nm.

Results and Discussion
We first confirmed that the dual-resonance effect is still effective on the tapered fiber. Figure 5 shows the transmission spectra of the BLPFGs with a cladding diameter of 121 µm and periods of 568.0 µm, 566.0 µm, 564.0 µm, and 562.0 µm. Each spectrum was filtered by FFT to remove the slight interference of LP 01 and LP 11 caused by the fusion of SMF and TMF. There were two resonance dips when the period was 568.0 µm. As the grating period decreases, the two resonant dips approach each other gradually, and the bridge between the two dips decreases. When the period was reduced to 562.0 µm, the two dips combined into a broadband dip and the 15 dB bandwidth reached the maximum of 168.5 nm from 1319.7 nm to 1488.2 nm. Thus, as the period decreases, the 15 dB bandwidth increases. The experimental spectra show the same change trend as the simulation results in Figure 2b.
The origin dispersion turning point of the used fiber was located at 1474.0 nm. After tapering to 121 µm and 113 µm, we moved the dispersion turning point to a shorter wavelength and realized broadband mode conversion. Figure 6

Results and Discussion
We first confirmed that the dual-resonance effect is still effective on the taper Figure 5 shows the transmission spectra of the BLPFGs with a cladding diamet μm and periods of 568.0 μm, 566.0 μm, 564.0 μm, and 562.0 μm. Each spectrum tered by FFT to remove the slight interference of LP01 and LP11 caused by the f SMF and TMF. There were two resonance dips when the period was 568.0 μm grating period decreases, the two resonant dips approach each other gradually, bridge between the two dips decreases. When the period was reduced to 562.0 two dips combined into a broadband dip and the 15 dB bandwidth reached the m of 168.5 nm from 1319.7 nm to 1488.2 nm. Thus, as the period decreases, the 15 d width increases. The experimental spectra show the same change trend as the sim results in Figure 2b. The origin dispersion turning point of the used fiber was located at 1474.0 n tapering to 121 μm and 113 μm, we moved the dispersion turning point to a short length and realized broadband mode conversion. Figure 6 shows the conversion e  Table 3. We observed the output mode field of the fabricated BLPFG with a cladding diameter of 113 μm and a period of 528.0 μm. The obtained mode field by a CCD camera is shown in Figure 7. The clear two-lobed LP11 mode field indicates that the mode converter has a high conversion efficiency.  We observed the output mode field of the fabricated BLPFG with a cladding diameter of 113 µm and a period of 528.0 µm. The obtained mode field by a CCD camera is shown in Figure 7. The clear two-lobed LP 11 mode field indicates that the mode converter has a high conversion efficiency.

Conclusions
In summary, we proposed an ultra-broadband BLPFG mode converter based on the dual-resonance coupling effect. By changing the fiber cladding diameter, we moved the dispersion conversion point from 1474.0 nm to 1404.0 nm and 1303.4 nm and achieved working bandwidths of 168.5 nm and 148.8 nm, respectively. Since the diameter of the fiber taper can be made according to the parameters we need, the dispersion turning point can be adjusted to the desired position. Besides, the mode conversion bandwidth can be adjusted by changing the grating period. In this way, dual modulation of the dispersion turning point and the mode conversion bandwidth can be achieved. The proposed BLPFG broadens the scope of application of the dual-resonance effect and proposes a general method for widening the bandwidth of a fiber grating with tunable wavelength.