## Abstract

Recently, chirped and tilted fiber Bragg gratings (CTFBGs) have received great attention because they can realize suppression of stimulated Raman scattering (SRS) in high-power fiber lasers. In this study, the possible coupling between the core modes and cladding modes in CTFBGs inscribed in large-mode-area double-cladding fibers is investigated for the first time. Theoretical results show that the coupling between the LP_{11} mode and cladding modes would destroy the transmission spectra envelope only considering the coupling of LP_{01} for single-mode CTFBGs, which will degenerate the SRS suppression performance. This was confirmed experimentally by measuring the spectral response under different mode excitations. A reliable method is demonstrated to ease the LP_{11}-excitation-induced spectral deterioration by choosing an appropriate chirp rate for the inscription of CTFBGs, which is useful for improving the Raman suppression effect of large-mode-area double-cladding CTFBGs in high-power fiber lasers.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Over the last decade, high-power fiber lasers have seen rapid penetration into various fields, such as industrial manufacturing, scientific research, and medical applications [1], because of their more compact construction, higher energy conversion efficiency and relatively better beam quality compared with their bulk counterparts [1]. To date, 20-kW fiber amplifiers and 8-kW fiber oscillators have been reported [2,3]; however, barriers to achieving higher power still remain, where stimulated Raman scattering (SRS) is one of the key limitations to the power scaling of fiber lasers. To date, many methods have been used to mitigate SRS in fiber lasers [4–6]. However, conventional mitigations may cause related problem such as transverse mode instability (TMI) [7,8]. Utilization of broad-band rejection filters, such as long-period gratings (LPGs), is another feasible method to mitigate SRS [9]. However, LPGs have inherent cross-sensitivities to multiple external factors, such as, temperature, which would cause an excursion of the central wavelength and a decrease in the suppression ratio [10].

As a kind of wide-band spectral filter, chirped and tilted fiber Bragg gratings (CTFBGs) have received much attention in recent years because of their effective Raman suppression in high-power fiber lasers [11–17] along with advantages of an adjustable wavelength range and good thermostability [14]. In 2017, we first reported the application of single-mode CTFBGs for Raman filtering in fiber amplifiers [11]. Then, by inserting CTFBGs between the seed laser and the amplifier stage, we reported impressive suppression of SRS in 5 kW fiber amplifiers including tandem pumping architecture [12] and direct diode pumping architecture [13]. To further expand the applicability of CTFBGs in high-power fiber oscillators and amplifiers, we first studied the fabrication of CTFBGs in large-mode-area double-cladding fibers (LMA-DCFs) with a core/cladding diameter of 20/400 µm [14]. In 2019, Jiao et al. achieved effective Raman suppression in a kW-scale monolithic fiber oscillator using CTFBGs inscribed in 20/400 µm fibers [16]. Recently, Lin et al. reported Raman suppression within the gain fiber of high-power fiber oscillators, also using CTFBGs inscribed in 20/400 µm fibers [17]. Although CTFBGs inscribed in LMA-DCFs have been used for Raman suppression in high-power fiber laser systems, only fundamental mode coupling was considered in these studies. In actual operation of LMA-DCFs, such as 20/400 µm fibers, high-order mode like LP_{11} would inevitably be excited, which could affect the Raman suppression effect of CTFBGs. Therefore, studying high-order mode coupling in LMA-DCF CTFBGs is necessary, especially for fiber lasers with relatively poor beam quality.

In this paper, we study the possible coupling between the core guided LP_{11} mode and cladding modes in LMA-DCF CTFBGs for the first time. Simulations show that the transmission envelope is deteriorated considering the coupling of the LP_{11} mode to the cladding modes. It is theoretically and experimentally verified that the attenuation bandwidth becomes broader with a decrease in the central suppression ratio when high-order mode excited. Furthermore, a feasible method is proposed to promote the coupling efficiency of the CTFBGs by slightly adjusting the chirp rate according to the mode proportion of the input light, which has guiding significance for the property optimization of CTFBGs in Raman suppression of high-power fiber lasers.

## 2. Simulations

Before the formal spectral simulation, it is vital to compute two categories of databases, including the corresponding coupling constants and resonant wavelengths, both of which are co-decided by the features of the guided modes in the fiber and the structure of the selective grating. We consider the typical modality of optical fibers consisting of a core, a cladding, and surrounding air (Fig. 1(a)). Taking the standard structure of two commercial LMA-DCFs for reference, two core modes are allowed to transmit, including the LP_{01} (Fig. 1(b)) and LP_{11} (Fig. 1(c)) core modes. The corresponding structural configurations of the LMA-DCFs are listed in Table 1. In this model, the surrounding is air as LMA CTFBGs are often used without coating. Subsequently, the structure of a CTFBG is discussed in the first subsection.

#### 2.1 Structure of CTFBGs

As shown in Fig. 2, CTFBGs are similar in design to a non-uniform period with chirped fiber Bragg gratings (CFBGs), compared with which the grating plane in CTFBGs skews off the fiber axis at an angle of *θ*. Here the direction of the fiber axis is defined as the *z*-axis and the direction along which the refractive index (RI) originally alternates is defined as *z*’. Because of the tilted plane, the period of the grating along the *z*-axis is lengthened by:

*Λ*is the original period along the

_{g}*z*’ axis. In addition, considering the chirp introduced in the grating,

*Λ*can be derived as:

_{g}*Λ*

_{g}_{0}is the average period of the grating,

*F*is the chirp coefficient, and

*L*is the length of the grating.

Based on the uncoated structure of an LMA-DCF (Fig. 1(a)), the original RI modulation along the *z*’ axis can be expressed as:

*n*

_{1}is the RI of the fiber core,

*n*

_{2}is the RI of the fiber cladding,

*n*

_{3}is the RI of the surrounding atmosphere, Δ

*n*is the amplitude of the ac RI modulation and

*β*=2

_{Bragg}*π*/

*Λ*representing the Bragg wave vector of a CTFBG with a period

_{g}*Λ*along the

_{g}*z*’ axis.

To describe the modulation of the axial RI, a certain coordinate transformation with the original *x-z* axis rotated by an angle of *θ* to *x*′-*z*′ axis (Fig. 2) can be introduced as follows:

Inserting the coordinate transformation into Eq. (3), the modulation of RI in the grating can be rewritten as:

**Δ**

*n**represents the alternating part of RI. The expression of index modulation is used in computing the overlap integral in the 4*

_{ac}^{th}subsection. In the computing, CTFBGs inscribed in two categories of LMA-DCFs are considered, of which the parameters (Table 2) were set according to the actual fabrication in our laboratory.

#### 2.2 Cladding mode field distribution

The mode field confined by the fiber structure can be precisely derived by solving the vector wave equation, the solved results of which, however, can be rather complex. As a matter of fact, many reports have shown that the linearly polarized (LP) approximation is also a reliable method for solving the coupling between guided modes and cladding modes with both short- and long- period gratings [18–22]. Here, a two-layer model is used to calculate the mode distribution of cladding modes [20], ignoring the fiber core. Although LP simplification was adopted, the complete calculation of the spectral response of CTFBGs still requires a tremendous workload when a large number of cladding modes need to be included, let alone the calculations involving the mutual coupling between core modes because of the LMA condition.

Because of the small coverage of the RI modulation (only inside the fiber core), the core modes can only couple to some specific groups of cladding modes [18,22] with the modes that have a weak distribution in the core, LP_{24} and LP_{34} (Fig. 3(c) and (d)), uncoupled. In the simulation, two groups of cladding modes, LP_{0n} and LP_{1n}, both of which have a strong field intensity distribution in the core, are considered. Typical examples of these two types of mode field distributions are shown in Fig. 3(a) and (b).

The tilt design of the grating plane in the CTFBG is also a kind of anisotropic structure, that affects the mode polarization state. However, considering the small-angle tilt of the grating discussed here, the change in the polarization state of the propagating modes is neglected in the later simulation.

#### 2.3 Resonant wavelengths of CTFBGs

Based on the phase-match condition [23], the corresponding resonant wavelengths of mode coupling can be obtained as

*Λ**is a matrix with elements representing each resonant wavelength of the different kinds of mode coupling mentioned above, the superscript ‘core’ and ‘cladding’ of the matrix elements respectively represent the coupling of core modes and mutual coupling between core and cladding modes, the subscript represents the order of the coupling modes,*

_{res}*n*is the effective index of the

_{clad,i}*i*

^{th}cladding mode, and

*n*

_{core,}_{01}and

*n*

_{core,}_{11}represent the effective index of the LP

_{01}and LP

_{11}core modes, respectively. Furthermore, based on Eqs. (2) and (6), the bandwidth of each coupling resonance can also be expressed in matrix form as:

Based on the formula deduced above, two important aspects that primarily affect the resonant wavelength and corresponding bandwidth should be manifested. On the one hand, the structural parameters of the CTFBG play a large part in determining the resonant wavelength. Acting as connecting thread filling up the detuning in phase matching, the period *Λ* along the fiber axis, which is jointly dependent on the original period of the grating *Λ_{g}* and the tilt angle

*θ*, linearly influences the coupling wavelength

*λ*according to Eq. (6).The corresponding bandwidths of each resonance also show a linear growth with an increase in the grating period

_{res}*Λ*and chirp rate

_{g}*F*according to Eq. (7). On the other hand, it is conspicuous that the effective indices of both pairwise-coupling modes also have a large influence on the determination of individual resonant wavelengths, according to Eq. (6).

Actually, the resonant wavelength of one cladding mode is also proportional to the sum of effective indices (*n _{core}+n_{clad,i}*) of the coupling core mode and cladding mode. Therefore, with the LP

_{11}core mode considered, the resonant wavelength of the cladding modes would see a drift for the inter-mode dispersion (difference between

*n*

_{core,}_{01}and

*n*

_{core,}_{11}) between the LP

_{01}and LP

_{11}core modes. Figure 4(b) visually reveal the resonant wavelength drift of cladding modes for coupling with different core modes.

A shift in the resonance peak wavelength would also occur when the effective indices of cladding modes, which are extraordinarily sensitive to the temperature and RI of the ambient medium, vary with the change in external factors. Moreover, as the increasing inter-mode dispersion between higher-order neighboring modes showed in Fig. 4(a), a larger wavelength spacing between the adjacent resonances could be expected in the shorter wavelength direction, resulting in a series of changes in the envelope of transmission spectra under different tilted angles. Further discussion of the properties of the spectra of CTFBGs will be continued in the next subsection.

#### 2.4 Coupling constants and simulated spectra

Computing of coupling constant is another essential component in simulating the resonance of mode coupling, which can be used to represent the coupling strength of two modes, and coupling constant is mainly defined by the overlap integral of the pair-wise coupling mode field along with the RI modulation. The coupling constant *κ** _{l,m}* of the two arbitrary modes coupling with each other can be expressed as [19]:

*ε*(

*r*) = 2

*n*

_{1}

**Δ**

*n**represents the permittivity perturbation in the grating,*

_{ac}*ω*represents the light frequency and

*e**and*

_{l}*****

*e**represent the normalized electric field distributions of the*

_{m}*l*

^{th}and

*m*

^{th}guided modes, respectively.

Based on Eqs. (5) and (9), the coupling between two different core modes can be considered non-existent in the ideal fiber Bragg grating (FBG) if RI modulation is considered transversely uniform, because of the inconsistent symmetry of LP_{01} and LP_{11} (Fig. 5). Only the modes in the same symmetry can couples with each other. However, the tilted grating plane induces an inhomogeneous profile of cross RI (pink line in Fig. 5), which gives rise to a nonzero overlap integral between the fundamental mode and modes with odd symmetry (gray line in Fig. 5). Basing on Eq. (9), the self-coupling (coupling between the same modes transmitting in opposite directions) and mutual-coupling (coupling between different modes transmitting in opposite directions) constants of the core modes were all calculated as shown in Fig. 6(a), which shows the alternate change in *κ*_{01}, *κ*_{11} and *κ*_{01,11}, corresponding to self-couplings of LP_{01} and LP_{11} and mutual coupling between LP_{01} and LP_{11}, respectively, with increasing tilt angle (the parameters of the 15/130 μm fiber are used in this part of the simulation). Observing the alternation of coupling constants, an apparent conclusion can be drawn that both the self and mutual coupling of core modes would be dramatically weakened when the tilt angle increases to greater than 4°.

In Fig. 6(b), we show the drop-off of the normalized *κ*_{01} for various radii of the fiber core with the increase in tilt angle. The coupling constants with a larger radius show a steeper decrease as the tilt angle increases. In other words, for the CTFBGs fabricated in fibers with larger core sizes, the core mode coupling could be strongly suppressed even with a smaller tilt angle, which has already been repeatedly verified by detecting the reflectivity spectra of CTFBGs [14,16]. Therefore, in the latter spectral simulation, the couplings between the core modes are neglected.

The largest part of the database must be prepared in advance and lies in the calculation of mutual coupling constants that define the interaction strength between core modes and numerous cladding modes. It is further used in the coupled mode equation for spectral computation [24]. The following is an analysis of the evolution of coupling coefficients, along with the corresponding manifestation about the simulated transmission spectra. Figure 7 shows how the mutual coupling coefficients (parameters of 15/130 are still used), which includes four types of coupling (interaction between LP_{01} and LP_{0n}, LP_{01} and LP_{1n}, LP_{11} and LP_{0n}, and LP_{11} and LP_{1n}), varies with both the radial order *n* and tilt angle from 0 to 10 degrees. To simplify the simulation, only cases of one pure core mode excitation are considered.

By observing the alternate change in simulated coupling constants shown in Fig. 7, four main regularities can be summarized, matching up with some information which is embodied in the simulated spectra shown in Fig. 8. Firstly, for a larger tilt angle, the radial order of the cladding modes, which strongly couple with the core modes, becomes higher (Fig. 7), leading to a shift in the central wavelength (Fig. 8(a)–(d)), i.e., the larger the tilt angle, the higher the order of the strongly-coupled cladding mode and thus the shorter the central wavelength, because of the smaller effective index of coupled cladding modes. In addition, although the total number of cladding modes with obvious coupling constant remains unchanged while the tilt angle grows (Fig. 7), it still makes sense that, as displayed in Fig. 8(a)–(d), the spectral wavelength range broadens with increasing tilt angle because the wavelength spacing between the higher-order cladding modes would become larger (Fig. 4(b)). Moreover, for a small tilt angle close to zero, stronger coupling is seen in Fig. 7(a) and (d) because of the good field overlap between modes with the same symmetry (LP_{01} and LP_{0n} are both axis-symmetric while LP_{11} and LP_{1n} are circuit-symmetric), in contrast with a large tilt angle condition where both core modes exhibit a weaker coupling with axis-symmetric cladding modes (Fig. 7(a) and (c)), which is attributed to the severe axial asymmetry of the RI profile.

The last point to mention is practically relevant to the application of CTFBGs in SRS filtering. As shown in Fig. 7(a)–(d), all sets of coupling coefficients (either with LP_{01} or LP_{11} mode excited) follow an oscillational change with the increasing radial order *n* of the cladding modes. However, different change rule appears when it comes to the respective radial orders of the cladding modes strongly coupled with the core-guided LP_{01} mode (Fig. 7(a) and (b)) and LP_{11} mode (Fig. 7(c) and (d)). The radial orders of cladding modes strongly coupled with LP_{01} mode are discontinuous compared with the successive radial orders of those strongly coupled with LP_{11} mode. This different coupling rule is further shown in the spectra of the CTFBG in Fig. 8(e), where the red curve presents two distinct dips. The middle hillock between two dips is just caused by the weaking coupling between LP_{11} mode and cladding modes with radial orders in the middle-disconnected part. For 1134 nm (the central SRS wavelength of a 1080-nm fiber laser), the suppression ratio has a gap of 9.08 dB between the LP_{01} coupling (blue spectrum) and LP_{11} coupling (red spectrum), as the 1134 nm is exactly located in the saddle bewteen two dips in the red curve. In other words, it primarily reveals that some negative influence would be caused to the filtering effect of CTFBGs because of the excited LP_{11} core mode. This conclusion is further supported by the corresponding experiments, the results of which would be discussed and compared with the simulation in the next section.

## 3. Experimental results and discussion

#### 3.1 Experimental setup and multimode excitation

The experimental setup shown in Fig. 9 is constructed to realize obvious excitation of multiple core modes in LMA-DCF, where three functional elements are combined, i.e., the mode excitation controller, the signal generator and the receiver. The mode excitation controller consists of a laptop and fusion splicer, the two sides of which fixed the pigtails of the single-mode fiber (HI1060) and LMA-DCF with clamps. By operating the manual controller of the fusion splicer through the laptop, the incident position on the LMA-DCF with respect to the HI1060 can be achieved with high accuracy (Fig. 9). Changing transverse offset between two fiber cores is a common method for excitation of higher-order mode in fibers [25,26].

The CTFBGs written in LMA-DCF were realized by the UV exposure system with a phase mask rotated by a certain angle around the axis of the inscribed light beam [14]. Two CTFBGs (length of 35 mm) were inscribed on 15/130 and 20/400 μm fibers using a phase mask with periods of 785.8 nm and 788.8 nm, respectively. Rotated by 4° and 6°, the corresponding chirp rates of the phase mask are 0.8 nm/cm and 4 nm/cm, respectively.

To analyze the proportion of the excited core modes quantitatively, the multimode interference effect [27] is introduced theoretically. The power coupling coefficient *η _{μ}* of the

*μ*

^{th}core mode can be obtained using an overlap integral:

Here, ** e_{i}**(

*r*,

*d*) and

**represent the profiles of the incident field and the**

*e*_{ν}*ν*

^{th}core mode respectively and

*d*is the offset between the centers of the incident field and the receiving fiber core. Based on [Eq. (10)], the variation curves of mode excitation which change with the offset

*d*, are computed and displayed in Fig. 10 (the incident field

**(**

*e*_{i}*r*,

*d*) was simulated using the parameters of the HI1060 fiber with the a core diameter of 5.3 μm, cladding diameter of 125 μm and numerical aperture of 0.14). When the two core modes see an equivalent excitation, the offset values are 5.0 μm and 5.6 μm, respectively, for 15/130 and 20/400 μm LMA-DCFs.

#### 3.2 Spectral results of CTFBGs

Using the experimental setup shown in Fig. 9, the transmission spectra of both CTFBGs inscribed in 15/130 μm and 20/400 μm fiber with various mode proportions were measured. In Fig. 11(a), the measured spectrum (blue line) of the 15/130 μm CTFBG with only fundamental mode excitation is displayed. With the tilt angle of 4°, the corresponding coupling constants of the LP_{01} mode are again shown in Fig. 11(b) and (c), where cladding modes whose radial orders are between 14 and 33 show obviously stronger coupling than others. By comparing the phase-matching condition (Fig. 11(d) and (e)) with the measured spectrum, it is shown that the wavelength range matches well with the resonant wavelength range of the strongly coupled cladding modes.

Using the same grating parameters of the fabricated 15/130 μm CTFBG, the corresponding simulated spectrum (red dashed line in Fig. 11(a)) was calculated and showed a similar transmission envelope, with a slight red shift compared with the real spectra (blue solid line), which could be due to the fact that the effective indices obtained by the LP method are slightly larger than those of the degenerate vector modes and that the slightly larger tilt angle or smaller RI of core in real can be influential as well. In addition, the 3 dB bandwidth of the simulated spectrum (8.43 nm) is smaller than the measured one (8.96 nm), mostly caused by neglecting the cladding modes with higher angular orders (m > 1).

For a similar full fundamental mode excitation (Fig. 12(c)) in the 20/400 μm fiber, however, it differs slightly. While the red shift is still shown in Fig. 12(c), the spectral range of the simulated spectrum shows a wider bandwidth (10.01 nm) than the measured one (9.82 nm). This difference can be due to setting a higher RI modulation, a longer grating length or a larger chirp rate in the simulation with respect to the real case.

Theoretically, an equivalent excitation of two core modes can be realized by carefully adjusting the transverse mismatch to the simulated results shown in Fig. 8. The corresponding transmission spectra of 15/130 μm and 20/400 μm CTFBGs are shown in Fig. 12(a) and (d). Both the measured transmission spectra of the two types of CTFBGs show a wider 3 dB bandwidth than the spectra of pure LP_{01} excitation because of the excitation of multiple core modes. The spectra of the case with a larger transverse offset between the single-mode fiber and 15/130 μm fiber are also displayed in Fig. 12(b). An even broader spectral range is shown by the measured result (blue line) as the proportion of LP_{11} mode increases. The simulated result (red line) reveals a shallower dip in the shorter wavelength range, as many kinds of cladding modes that can couple with core modes are neglected in the simulation and the impact of these neglected pieces obviously plays a larger part in the shorter wavelength range because of the more severe modal dispersion (Fig. 4(a)). Moreover, with an increase of the higher-order mode proportion, the central suppression ratio (at 1134 nm) of the fabricated 15/130 μm CTFBG shows a faster decrease than the anticipated, decreasing from 9.92 dB to 3.35 dB, which is unfavorable for SRS suppression. In the next subsection, a theoretically feasible method to solve this problem is further discussed.

#### 3.3 Optimized design of CTFBGs

Improving the spectral response is beneficial to the optimization of the CTFBG property for SRS suppression. First, as a natural thought, we attempted to realize the target of eliminating the bad effect (LP_{11} mode induce) by moving the saddle point of the curve representing LP_{11} coupling out of the SRS wavelength. This thought can be achieved by using a larger tilt angle. However, as the tilt angle increased, the suppression ratio of the overall wavelength would decrease (Fig. 8(a) and (b)) because of the weaker coupling of all cladding modes. Therefore, optimization of the other CTFBG parameter is discussed in this section.

In Fig. 13(a) and (b), the variation of the simulated transmission spectra that evolve with the changing chirp rate is displayed. As the chirp rate increases, the suppression ratio (at 1134 nm) for the LP_{01} mode is decreases because of the expansion of the spectral energy distribution. For LP_{11} mode coupling, the suppression ratio is a competition between the broadening of the spectral range and the superposition of the original two transmission dips. To further judge the effect the chirp rate could have on improving SRS filtering when two core modes were excited, an increase in the suppression ratio (set the suppression ratio of 0.4 nm/cm as the basis point of zero growth) on conditions of different proportions of the LP_{01} core mode excitation for chirp rate from 0.40 to 0.75 nm/cm is shown in Fig. 13(c).

Suggestions can then be drawn from the simulated results of various chirp rates: when the LP_{01} core mode takes a proportion of greater than 65%, it is better to retain a weak chirp rate at 0.4 nm/cm, as the total filtering effect is on a declining curve, while the chirp rate gradually increases. As the proportion is inferior to 65%, there would be an optimal chirp rate that can balance the reduction of suppression ratio for the LP_{01} core mode and the increase in that for the LP_{11} core mode. For the proportion between 45% and 65%, the optimal chirp rate is 0.6 nm/cm. When the LP_{01} mode shares less than 45% of the total field, the ideal chirp rate should be 0.65 nm/cm.

Determining the best parameters of the CTFBG for SRS filtering would be a time-consuming process of endless searching, which may nevertheless be meaningful for expanding the application of CTFBGs in high-power fiber systems. In the near future, further testing of the simulated results will be carried out experimentally.

## 4. Conclusions

In conclusion, considering the possible motivation of multiple core modes for LMA fibers, we investigated the coupling characteristics between the LP_{11} core mode and cladding modes in a CTFBG. By calculating the coupling constants and resonant wavelengths, we simulated the transmission spectra of CTFBGs with different parameters and compared the results of LP_{01} and LP_{11} coupling. With respect to the condition of LP_{01} core mode coupling, simulations show that LP_{11} coupling can cause spectral widening and a decrease in the suppression ratio for the SRS wavelength, which is further confirmed by the measured spectra with the motivated field of different mode proportions. To find the proper parameters and improve the performance of CTFBGs in the work environment of multiple core mode excitations, transmission spectra under different chirp rates is simulated. Comparing the enhancement of the SRS filtering effect, the optimal chirp rate was chosen under various ratios of mode components. In the future, by finding better parameters and consistently promoting the property of CTFBGs, a superior SRS suppression effect would be expected, which will be a huge advancement for further power scaling of high-power fiber systems.

## Funding

State Key Laboratory of Power Systems (SKL-2020-ZR05); National Natural Science Foundation of China (11974427, 12004431); Natural Science Foundation of Hunan Province (2019JJ20023).

## Disclosures

The authors declare no conflicts of interest.

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