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Abstract
We demonstrate the fabrication of over-coupled long-period fiber gratings (LPFGs) in the 1.55-µm and 2-µm wavebands enabling broadband linearly polarized LP11 mode conversion using a CO2 laser. The birefringence of the fiber is caused by on one side laser exposure and increases with the increase of refractive index modulation depth, which realizes the conversion of linearly polarized modes. The mode conversion bandwidth can be significantly increased by using the over-coupled LPFG. The 10-dB bandwidth of the LPFGs with |κ|L values of π/2, 3π/2, and 5π/2 are 33.04, 80.84, and 114.08 nm at 1.55 µm waveband, respectively. The maximum bandwidth of the over-coupled LPFG is 3.79 times higher than that of conventional LPFG. The operating wavelength of the mode converter can be extended to 2.0 µm wavebands and the maximum 10-dB bandwidth reaches 161.32 nm. The proposed broadband linearly polarized mode converters could have potential application in the fields of mode division multiplexing systems, fiber laser systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Space division multiplexing (SDM) technology is one of the potential technologies of next-generation optical fiber communications, which is to increase the optical fiber communication capacity by increasing the spatial channel of data transmission [1,2]. Mode division multiplexing (MDM) is a type of SDM that utilizes few-mode fiber (FMF) to support multiple fiber modes to increase independent data channels [3,4]. To transmit independent data over fiber mode, mode converters are required to convert the fundamental mode to high-order core modes that can be guided in FMF. Implementations of mode converters have been reported, such as spatial light modulators (SLM) [5,6], mode-selected couplers [7], fiber Bragg gratings (FBGs) [8,9], photonic lanterns [10], long-period fiber gratings (LPFGs) [11,12]. Benefiting from the advantages of LPFGs such as small size, easy fabrication, high flexibility, and good optical fiber compatibility, LPFG-based mode converters have attracted great attention in recent years. The vector modes of the same mode group have the same propagation constants in circular-core FMF, so the state of polarization of the mode converters based on vector modes is unstable. The linearly polarized (LP) mode is the linear combination of the vector modes, which is more commonly used because it is more stable and easy to identify. Recently, the reported methods for realizing LP mode converter include the introduction of asymmetric refractive index modulation in circular-core FMF [13,14], and the fabrication of the LPFGs in high birefringence fibers such as elliptical core fibers [15–17], polarization-maintaining fibers [18,19]. Nevertheless, the bandwidth of the conventional LPFGs as mode converters is usually only tens of nanometers.
To fully exploit the capabilities of MDM, the incorporation of wavelength division multiplexing (WDM) techniques further requires mode converters with wider bandwidths. Many methods have been proposed to realize broadband mode conversion. One is to insert multiple phase shifts in the grating structure to achieve broadband mode conversion [20–22], K. S. Chiang et al. propose a long-period grating structure with apodization lengths, which introduces π phase shifts in the grating, enabling broadband conversion of LP01 to LP11a and LP11b [20]. The bandwidth of the mode converter can also be extended by designing the chirp parameters and the number of cascaded gratings [23]. Although these methods have achieved broadband mode converters, the large number of design parameters, complex fabrication processes, and large grating lengths limit their application. Another way to realize broadband mode conversion is to inscribe the LPFGs at the dispersion turning point (DTP) of the phase matching curve [24–26]. C. Jiang et al. realized broadband LP mode conversion at 1.0 µm and 1.5 µm by fabricating the LPFGs operating at DTP in two kinds of few-mode polarization-maintaining fibers [26]. However, the fabrication of DTP-LPFG requires specially designed few mode fiber, and the LPFG at the DTP is also sensitive to changes in the grating period and the external environment. The resonance wavelength of the broadband bandwidth is fixed near the DTP, so the tunable range is small.
In this paper, we theoretically and experimentally investigated the conversion of linearly polarized modes in a few-mode fiber by controlling the refractive index modulation depth of the LPFGs. A broadband mode converter is achieved using over-coupled LPFG. The 10 dB bandwidth of the over-coupled LPFGs with |κ|L values of 3π/2 and 5π/2 are 80.84, and 114.08 nm at 1.55 µm waveband, respectively. Finally, the operated wavelength of the mode converter can be extended to 2.0 µm wavebands with maximum 10 dB bandwidth of 161.32 nm. The bandwidth of the proposed mode converter can be further improved by combining other bandwidth extension mechanisms, which could find applications in the few-mode fiber-based communication and optical fiber laser systems.
2. Principle and simulation
The fiber used in the experiments is a step-index FMF (four-mode fiber, OFS), the fiber with core/cladding diameter of 25 µm/125 µm, and the index difference between the core and cladding is 0.005. We performed numerical simulations based on the finite-element method (FEM) with commercial software (COMSOL Multiphysics). The mode coupling can be obtained by introducing periodic refractive index modulation with the FMF. We investigate the effect of refractive index modulation (Δn) on the LP11 mode intensity and state of polarization. Since the LPFG is inscribed by one side of CO2-laser exposure, a graded refractive index modulation model is introduced on the fiber cross-section. Figure 1 shows the mode field distribution of LP11 mode under different refractive index modulation depths. Without refractive index modulation Δn = 0, the fiber supports four eigenmodes, including $T{M_{01}}$, $HE_{21}^{even}$, $HE_{21}^{odd}$, $T{E_{01}}$ mode. As the refractive index modulation depth increases, the annular eigenmodes gradually evolve into two lobes. When Δn = 2 × 10−4, ${\; }HE_{21}^{even},\; T{E_{01}}$ have been converted into two lobes, generating $\textrm{LP}_{11b}^y$, $\textrm{LP}_{11a}^x$. When Δn reaches 6 × 10−4, the four eigenmodes are completely transformed into linearly polarized (LP) modes, corresponding to $LP_{11a}^x,\; LP_{11b}^y,$ $LP_{11b}^x,\; LP_{11a}^y$. Figure 2(a) shows the effective refractive index of the four eigenmodes of the LP11 mode decreases as the refractive index modulation depth increases. When the refractive index modulation increases from 0 to 6 × 10−4, the effective refractive index of the eigenmodes is reduced by 9.0 × 10−7, 4.9 × 10−7, 5.0 × 10−6, 5.4 × 10−6, respectively. Due to the birefringence, the same mode has different propagation constants in the x and y polarization direction, which can be reflected in different effective refractive indexes in the x and y polarization [27]. According to $\Delta {n_B} = n_{neff}^x - n_{neff}^y$, where $\Delta {n_B}$ is the birefringence of the mode, and $n_{neff}^x,\; n_{neff}^y$ is the mode effective refractive index of the x and y polarization. As shown in Fig. 2(b), the birefringence of both LP11a and LP11b modes increases with the depth of index modulation. The large birefringence induced by asymmetric refractive index modulation of CO2 laser. The refractive index difference between the eigenmodes of the LP11 mode increases gradually with the increase of the refraction index modulation. Therefore, by increasing the refractive index modulation depth of the LPFG, the mode conversion between the fundamental mode and high-order LP mode can be achieved.
Fig. 2. (a) The effective refractive index of the four eigenmodes of the LP11 mode as a function of the refractive index modulation. (b) Birefringence of LP11a and LP11b as a function of the refractive index modulation.
We further theoretically analyze the effect of refractive index modulation depth on the bandwidth of the LPFGs. When the coupling of the two modes is phase-mismatched, the co-directional mode coupling efficiency η obtained by the LPFG can be defined as [28]:
We further calculate the coupling bandwidth for different values of |κ|L, by fixing the resonance wavelength λ0 and the length L. According to Eq. (3), the bandwidth of mode coupling depends on the m|κ|. If |κ|L is taken as χ1 and χ2 (χ2 > χ1), respectively, the ratio of the coupling bandwidth can be expressed as:
Fig. 3. Functional relationship between coupling efficiency and phase mismatch at different |κ|L values, assuming a fixed coupling length L.
Because the coupling length of the LPFG is a fixed value, |κ2|>|κ1| when χ2 > χ1. Although the value of m decreases, m2|κ2| is greater than m1|κ1|. Next, the ratios of the 10 dB bandwidth of the LPFGs with |κ|L of 3π/2 and 5π/2 to the 10 dB bandwidth of LPFG with |κ|L of π/2 are calculated to be 2.61 and 3.79, respectively. Therefore, we can effectively increase the coupling bandwidth by increasing the value of |κ|L. At a constant L, the value of |κ|L can be increased by controlling the refractive index modulation depth, and a wide coupling bandwidth will achieve at large |κ|L. Therefore, the over-coupled LPFGs with |κ|L > π/2 should be adopted.
Based on the finite element analysis method, we use the commercial software COMSOL to calculate the effective refractive index of the LP01 and LP11 modes. The effective index of LP01 and LP11 modes in the wavelength range from 1100 to 2200 nm is shown in Fig. 4(a). According to the phase matching conditions of the LPFG, we calculated the phase-matching curve of the FMF-LPFG for LP01-LP11 mode conversion, as shown in Fig. 4(b). The transmission spectra of LPFG inscribed in the FMF were calculated by Rsoft software based on beam propagation method. The period and length of the LPFG are set to 1790µm and 35.8 mm, respectively. By adjusting the value of the refractive index modulation (Δn) to change the |κ|L value of the LPFGs, the evolution of the transmission spectra of the LPFG can be observed. Figure 5 shows the evolution of the transmission spectra of LPFG with increasing refractive index modulation from 2 × 10−4 to 1.4 × 10−3. When Δn increases from 2 × 10−4 to 4 × 10−4, the |κ|L increases from π/2 to π, and the coupling efficiency of the grating decreases. When Δn is 4 × 10−4 (i.e., the |κ|L is about π), the contrast of the resonance wavelength of the LPFG almost disappears. When the Δn increases from 4 × 10−4 to 6 × 10−4, i.e., the |κ|L increases from π to 3π/2, and the LPFG coupling efficiency gradually increases. The LPFG has the maximum grating contrast again when the |κ|L is close to 3π/2 (i.e. Δn = 6 × 10−4). The 10 dB bandwidth of the LPFG is 77.20 nm, which is 2.52 times the 10 dB bandwidth of the LPFG when |κ|L is π/2. The difference in the resonance wavelength may be due to the wavelength shift caused by the increase of the refractive index modulation and the increase of the bandwidth. As the modulation depth Δn increases to 10−3, the |κ|L approaches 5π/2, and the grating contrast increases to be the maximum again. But increasing the Δn to 1.4 × 10−3, i.e., the |κ|L > 5π/2, the extra loss of the LPFG increases greatly, which is unfavorable to the application of the LPFG.
Fig. 4. (a) The effective index of LP01 and LP11 modes in the wavelength range from 1100 to 2200 nm. (b)The dependence of the grating periods and the resonance wavelengths according to the phase matching conditions.
Fig. 5. The evolution of the transmission spectra of LPFG with increasing refractive index modulation depth from 2 × 10−4 to 1.4 × 10−3.
To investigate the 10 dB bandwidth of the LPFG with different |κ|L in 1.55 µm and 2.0 µm waveband, the transmission spectra of the gratings with periods from 1510-1850µm are calculated by Rsoft, as shown in Fig. 6. When |κ|L is π/2, the 10 dB bandwidth of the LPFGs are 53.0 nm, 50.0 nm, 46.7 nm, 44.5 nm, 42.0 nm, 37.7 nm, 33.6 nm, and 30.6 nm respectively, as shown in Fig. 5(a). The 10 dB bandwidth decreases with the increase of the period, which may be due to the resonance wavelength of the LPFG with a large period being in the short wave. Figure 5(b) displays that the 10 dB bandwidth of the LPFGs are 113.0 nm, 117.1 nm, 107.6 nm, 99.6 nm, 87.4 nm, 84.2 nm, 77.2 nm, and 72.6 nm when |κ|L is 3π/2. The 10 dB bandwidth is expanded by about 2.3 times, which is consistent with the predicted trend of Eq. (4).
Fig. 6. The transmission spectra of the LPFGs with period from 1510-1850µm, (a) |κ|L is π/2; (b) |κ|L is 3π/2.
3. Experimental results and discussions
In the experiment, we fabricate the over-coupled FMF-LPFG by point-to-point writing technology using a CO2 laser [15]. The super-continuum broadband light source (NKT Photonics) and an optical spectrum analyzer (AQ6375, YOKOGAWA) are used to monitor the evolution of over-coupled FMF-LPFG during the fabrication. Both ends of the few-mode fiber are connected to the single-mode fiber. The period and number of periods were set to 1790µm and 20, respectively. The coupling length L is the length of the LPFG, which is 35.8 mm.
The refractive index modulation depth can be controlled by adjusting the number of laser scanning cycles. The method of writing dynamic analysis was proposed to study the physical mechanism of the CO2 laser inscribed LPFGs [30]. By using lower energy density of laser exposure, the fabrication of the LPFG can be divided into several tens of laser scanning cycles. A scanning cycle was completed when the number of the LPFG periods required was attained. By increasing the number of laser scanning cycles, the refractive index modulation in the FMF increases. The scanning cycle could be repeated multiple times at the same position. By monitoring the transmission spectra of the LPFG during fabrication, the coupling efficiency (|κ|L) can be controlled. As the number of scanning cycles increased, the value of |κ|L gradually increased from 0 to 5π/2. Figure 7(a) a shows the evolution of the grating contrast and wavelength of the LPFG with the number of scanning cycles. The grating contrast of the LPFG exhibited a sinusoidal change. The three peaks in the contrast change process correspond to the |κ|L values of π/2, 3π/2, and 5π/2, respectively. Figure 7(b) shows the transmission spectra of the LPFGs at the |κ|L values of π/2, 3π/2, and 5π/2, respectively. The coupling efficiency of the LPFGs can reach more than 99%. The extra insertion loss of the LPFGs increases with the value of |κ|L. When |κ|L is 5π/2, the maximum extra insertion loss is lower than 0.35 dB. The 10 dB bandwidth of the LPFGs increases as the |κ|L value increases, which are 33.04 nm, 80.84 nm, and 114.08 nm, respectively. The bandwidth at |κ|L values of 3π/2, and 5π/2 is 2.30 and 3.45 times larger than at |κ|L values of π/2. Table 1 shows the calculated ratio of 10 dB bandwidth when the |κ|L values of π/2, 3π/2, and 5π/2 according to Eq. (4). The experimental and theoretical ratio of the 10 dB bandwidth between the over-coupled LPFG and conventional LPFG is consistent with each other, confirming that we can achieve broadband mode conversion by inscribing the over-coupled LPFGs. The experimental value is slightly smaller than the theoretical value, which may be due to measurement differences caused by the interference of the transmission spectrum in the experiment.
Fig. 7. (a) The evolution of the grating contrast and wavelength of the LPFG with the number of scanning cycles; (b) the transmission spectra of the LPFGs at the |κ|L value of π/2, 3π/2, and 5π/2, respectively.
Table 1. The ratio of 10 dB bandwidth when the |κ|L values of π/2, 3π/2, and 5π/2
To verify that the over-coupled LPFGs realized mode conversion from LP01 mode to linearly polarized LP11 mode, a tunable laser (81600B, Agilent) and CCD (InGaAs, Model C10633-23, Hamamatsu Photonics) are used to monitor the mode field intensity distribution of the over-coupled LPFGs [25]. Figure 8(a) shows the mode field intensity distribution for the LPFG ranging from 1510 nm to 1580 nm when |κ|L is 3π/2. The clear LP11a mode intensity distribution demonstrates that the mode conversion efficiency is very high and has broadband characteristics. A polarizer was further placed in front of the CCD, and the polarizer was rotated at 45° intervals to monitor the change of the LP11a mode field intensity at 1550 nm. In Fig. 8(b), the mode field intensity distribution has the phenomenon of alternating bright and dark, but the direction of the mode lobe does not rotate, indicating that LP11a mode is generated. When |κ|L is 5π/2, the mode field intensity distribution of LPFG ranges from 1540 nm to 1640 nm, as shown in Fig. 9. Compared with the mode intensity distribution when |κ|L is 3π/2, the size consistency of the two lobes of LP11a mode decreases when |κ|L is 5π/2, which may be due to the increased asymmetry of the fiber.
Fig. 8. (a) The mode field intensity distribution for the LPFG ranges from 1510 nm to 1580 nm when |κ|L is 3π/2; (b) the pattern changes with the rotation of the polarizer at 1550 nm.
Fig. 9. The mode field intensity distribution for the LPFG ranges from 1540 nm to 1640 nm when |κ|L is 5π/2.
We investigated the polarization-dependent loss (PDL) characteristics of the LPFGs at different |κ|L values by an optical vector analyzer (OVA5000, Luna). Figure 10 displays the PDL characteristics of the LPFGs. When |κ|L is π/2, 3π/2, and 5π/2, the maximum PDL of the LPFGs are 2.55 dB, 6.20 dB, and 11.67 dB, respectively. A larger |κ|L value requires an increase in the number of laser scanning cycles, thereby increasing the asymmetry of the fiber due to one side CO2 laser exposure, resulting in a rise in the PDL of the LPFG. This may also be the reason for the asymmetric increase of the two lobes of the LP11a mode when |κ|L is 5π/2.
To combine with wavelength division multiplexing technology, the over-coupled LPFGs with different periods in the 1.55 - 2.0 µm band to improve the tunable range of the broadband linearly polarized mode converter. The LPFGs with periods of 1820µm, 1790µm, 1750µm, 1580 µm, and 1520 µm were fabricated using CO2 laser, respectively. Figure 11(a) shows the transmission spectra of the LPFGs when |κ|L is π/2. The 10 dB bandwidths were measured to be 31.44 nm, 33.05 nm, 39.80 nm, 54.44 nm, and 54.13 nm, respectively. When |κ|L is 3π/2, the transmission spectra of the over-coupled LPFGs are shown in Fig. 11(b), and the 10 dB bandwidths of these LPFGs are 73.67 nm, 80.84 nm, 116.80 nm, 161.32 nm, and 143.14 nm, respectively. Figure 12 shows the experimental and simulated dependence of 10 dB bandwidth on different grating periods. The experimental results indicate that the grating bandwidth can be effectively improved by increasing |κ|L, and the bandwidth decreases with the increase of the period, which is consistent with the simulation. The difference between the experimental and simulated 10-dB bandwidth may be caused by the accuracy of the LPFG parameters during the fabrication process and the simulation.
Fig. 11. The transmission spectra of the LPFGs with periods of 1820µm, 1790µm, 1750µm, 1580 µm, and 1520 µm (a) |κ|L is π/2; (b) |κ|L is 3π/2.
4. Conclusions
We propose broadband linearly polarized mode converters based on the over-coupled LPFGs in 1.55 µm and 2.0 µm wavebands. The LPFGs with |κ|L values of π/2, 3π/2, and 5π/2 are fabricated by controlling the number scanning of the laser. Compared with the 10 dB bandwidth of the LPFGs with the resonance wavelength of 1.55 µm, the over-coupled LPFGs are 2.44 times and 3.45 times larger than the conventional LPFG, respectively. By inscribing over-coupled LPFGs (|κ|L = 3π/2) with different periods, a wide range of tuning can be achieved in the 1.55-2.0 µm band. This simple and flexible fabrication method enables the broadband and tunable linearly polarized mode converter to have better application potential in broadband MDM, wavelength-tunable fiber lasers, and other fields.
Funding
National Natural Science Foundation of China (61875117, 62075124).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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