1. Introduction

Few-mode (FM) fibers have recently attracted renewed interest for their applications in optical communication and sensing, sparked mainly by the development of mode-division-multiplexing (MDM) technique. In principle, MDM is based on the utilization of individual guided mode as distinct multiplexing channel, which dramatically increases the capacity of a single optical fiber [1,2]. Moreover, extension of the concept of MDM to optical fiber sensing leads to simultaneous multi-parameter sensing [3,4], since different fiber modes respond differently to the ambient environment. One of the major challenges for FM fiber-based applications in practice is how to selectively excite the desired fiber modes, owing to the presence of intermodal coupling which can significantly increase the data processing complexity [5]. To date, numerous mode control techniques such as using photonic lanterns [6,7], phase plates [1,2], computer-generated holography [8,9], and fiber Bragg gratings (FBGs) [10,11] have been successfully demonstrated. The latter, as being inscribed within the fiber, provides the system with additional stability and integrity.

FBGs inscribed into FM fibers with phase masks, either by ultraviolet (UV) irradiation [12] or by femtosecond (FS) lasers [13], have been reported previously as mode selectors in MDM-based communication [14,15] and sensing [16,17] systems, as well as in fiber laser applications [18–20]. However, the capacity of those aforementioned FBG-based systems, where one entire linearly-polarized (LP) mode group is utilized as only one multiplexing channel, can be further increased if one is able to lift and control the spatial degeneracy of the radially asymmetric LP modes (e.g., LP$_{11}$). Additionally, the mode selectivity of FM-FBGs (FBGs inscribed in FM fibers) in fiber lasers is usually achieved by an external polarization controller, which introduces to the lasing system additional instability. Consequently, it is often desirable to add onto the FM-FBGs the capabilities of optical field orientation manipulation and/or the polarization discrimination. The former capability can also benefit applications such as optical trapping and tweezers [21–23], where the need for optical field manipulation is frequently encountered. To achieve these novel properties, FS laser point-by-point (PbP) technology [24] is utilized to fabricate the FBGs in this article. First, unlike using the phase mask, PbP technique can significantly increase the inscription flexibility thanks to the use of objectives with high NA. Moreover, compared to the UV-based PbP inscription [25], the FS PbP technique takes additional advantages stemming from the unique physical mechanisms behind grating formation. For example, the underlying nonlinear absorption of the laser energy obviates the need of photosensitization [26], and the ultrashort pulse duration leads to highly localized refractive index modulation (RIM) as a result of a-few-femtosecond-long transfer of energy from the laser pulses to the electrons [27]. We note that although FS PbP inscription of FM- and multimode (MM)-FBGs have been reported in [28–30], only the reflectivies were characterized. Additionally, our work differs from the previous studies on off-axis FBGs [28,31–40], which were either limited to single-mode fibers or only the reflectivities were investigated.

In this work, we report a detailed experimental investigation of the off-axis FBGs inscribed in a step-index two-mode (TM) fiber by FS laser PbP technology, with multiple characteristics reported for the first time to our best knowledge. By means of polarization control and image acquisition of the grating-reflected signals, we study the dependence of the grating spectra on the offset position and on the direction of linear polarization (DoLP) of the incident light. It should be noted that we present our results on the basis of LP mode group but with polarization and spatial degeneracies considered, as a balance between the LP approximation and the hybrid vector eigenmodes [41]. Particularly, we report that the reflection of the LP$_{11}$ has strong capability of polarization discrimination, which differs in the horizontal and vertical off-axis FBGs: The much stronger peak reflectivity constantly corresponding to the polarization perpendicular to the damage-point-to-center line. We also find that the forward LP$_{01}$ couples backwardly with strong preference to one spatial LP$_{11}$ whose orientation determined by the DoLP of the incident light. This opens up the possibility to lift the spatial degeneracy and further to manipulate the spatial orientation of the radially asymmetric LP modes. This paper is organized as follows. Section 2 contains the two experimental setups of FBG fabrication and FBG characterization, and the experiment to determine the FBG orientation. In Section 3, we present detailed characterization results, as well as the qualitative analyses. Section 4 describes the conclusion and potential applications.

2. Experimental setups and FBG orientation

2.1 Experimental setup of FBG fabrication

The FBG samples are fabricated using the setup shown in Fig. 1(a). A Ti:Sapphire FS laser with a regenerative amplifier (Coherent Libra series) operating at $800$ nm is used in the fabrication. The laser emits a transversely Gaussian shaped pulse chain with beam waist of $3$ mm, pulse width of $\sim 100$ fs, and repetition rate ($f_R$) set at $500$ Hz. The power and polarization of the pulse is controlled by a pair of a half-wave plate ($\lambda /2$) and a linear polarizer. The pulse energy is tuned at $128$ nJ, which slightly exceeds the single-shot damage threshold of the glass [42] but not induces significant cracks. Afterwards, the pulse chain is tightly focused by an oil-immersion objective (Nikon, $100\times$, NA = $1.25$) into a step-index TM fiber (OFS, core diameter = $19$ µm, NA = $0.12$). The TM fiber is mounted on an assembled $5$-axis translation stage, which can linearly translate along $x$-, $y$-, and $z$-axes, and tilt around $x$- and $y$-axes, as depicted in the figure. To inscribe a periodic structure into the fiber, the translation stage is traversed in $z$ direction at a constant speed $v_z$, leading to an FBG with period $\Lambda$ determined by $v_z/f_R$. In our case, $\Lambda =1.076$ µm is set to inscribe a second-order FBG with the Bragg wavelength of the LP$_{01}$ ($\lambda _{01}$) designed at $1551.18$ nm. The standard deviation of the actual $\lambda _{01}$ of all samples fabricated is $0.09$ nm, subject to the random axial strains induced while manually loading the TM fiber. All FBGs are $5$ mm long. Meanwhile, the fabrication process is monitored in real time via a transmitted brightfield optical microscopy system which consists of a blue LED as illumination source, the same objective used for fabrication, and a CCD camera. More details of the fabrication setup can be found in [43]. In addition, as illustrated in the inset of Fig. 1(a), FBGs with vertical offsets are labeled as V-$x$, where $x$ refers to the distance off the fiber center in micrometers (µm). Similarly, H-$y$ stands for FBGs with horizontal offset distance of $y$ µm. Figures 1(b) and 1(c) display two representative microscope images of sample V-2 (FBG with 2 µm vertical offset). The dashed black lines mark the core boundaries, the red arrows indicate the propagation direction of the writing pulses, and the sizes of the grating structure are noted in white. As can be seen from the two images, albeit limited to resolution, the laser-induced structural change is featured as an ellipsoid with a central microvoid surrounded by a densified shell, and with the spatial long axis parallel to the direction of the writing pulses [44]. We notice that there appears to be additional RIM at the bottom of each damage point, which is likely caused by the spherical aberration at the objective/index fluid interface, since the index matching oil applied here matches the index of fiber ($\sim 1.45$) rather than the one required by the objective ($\sim 1.5$) [45].

 

Fig. 1. (a) Experimental setup for FBG fabrication. $\lambda /2$: half-wave plate. Inset depicts the cross section of fiber core region. V-$x$: the FBG with $x$ µm vertical offset; H-$y$: the FBG with $y$ µm horizontal offset. (b) Top view and (c) side view of the microscope images of sample V-2.

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2.2 Characterization of FBG orientation

Since the induced RIM is axi-asymmetric (as shown in Figs. 1(b) and 1(c)), it is important to know the orientation of the FBG before performing detailed characterization. To mark the FBG orientation, we launch into the fiber core laser light operating at $660$ nm, and record the resulting scattering patterns in a plane perpendicular to the fiber axis, as illustrated in Fig. 2(a). Two orthogonal straight lines are marked in the plane, with one parallel to the optical table. Figures 2(b)–2(i) show the recorded polar plots of the scattering patterns generated by the corresponding FBG samples. The two $\sim 180^\circ$-apart bright regions in the ring-like patterns are connected by dashed white lines, whose distances to the origin of the coordinate are indicated by the arrow pairs. It is noticeable in Figs. 2(c)–2(e) that the dashed white lines almost pass through the coordinate origin for horizontal-off-axis FBGs, whereas in the case of vertical offsets shown in Figs. 2(f)–2(i), there exist gaps between the white lines and the origin, and the gap length is proportional to the offset distance as expected. The FBG orientation is therefore determined such that the spatial short axis of the ellipsoidally shaped RIM is parallel to the dashed white line, as pictured in Fig. 2(i), where the compound of a blue ellipse and a white circle represents the damage point. In principle, one may link the FBG orientation to its scattering pattern by applying the theory of light scattering by spheroidal particles presented in [46]. Particularly, the TM fiber designed for c-band can support 17 LP modes at 660 nm, the random mixture of the 17 LP modes can be treated as almost-equally-excited transverse electric and transverse magnetic modes. Interpretation of the corresponding result in [46] to our setup leads to the two intensity maxima in the ring-shaped patterns being in the line parallel to the short axis of the ellipsoidal RIM. Similar scattering patterns were also observed in [47,48]

 

Fig. 2. (a) Schematic of the experimental setup to determine the FBG orientation. (b)–(i) Recorded scattering patterns of the corresponding FBG samples. The compound of a blue ellipse and a white circle in (i) represents the orientated damage point.

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2.3 Experimental setup of FBG characterization

The characterization of the FBG samples are carried out using the setup in Fig. 3. A tunable laser (New Focus, TLB-6728) generates a narrow-linewidth continuous wave at a desired wavelength $\lambda$. The laser beam is then coupled into free space through a collimator (L1) after passing through a fiber-based polarization controller (PC). The polarization of the collimated beam is tuned by a linear polarizer. Afterwards, the beam is focused into the TM fiber ($\sim 1$ m in length) with the FBG written at $\sim 20$ cm away from its proximal end. We keep the section between the fiber proximal end and the grating as straight as possible to reduce the intermodal coupling. The transmitted signals out of the fiber are collected by a photodiode (PD1) with $\varnothing 1$ mm active area located in close proximity to the fiber distal end. Prior to entering the fiber, the focused beam is partially reflected by a non-polarizing beam splitter (BS) to another PD (PD2). Signals from the PD1 and PD2 are acquired by a data acquisition (DAQ) card (National Instruments, NI USB-6211) and analyzed by a computer. To characterize the mode and the power of the light reflected by FBGs, we use an infrared camera (Goldeye, G-032) following the same BS to capture the intensity profiles of the reflected signals. The optical power is calculated by summing the values of all camera pixels. To be immune to the fluctuations of the laser power, here we define a normalized transmitted signal $\mathcal {T}$ and a normalized reflected signal $\mathcal {R}$ in arbitrary units respectively as

 

Fig. 3. Schematics of the experimental setup to characterize the FBG samples. PC: polarization controller; L1: collimator; L2: focusing lens; BS: beam splitter; PD: photodiode; DAQ: data acquisition.

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3. Characterization results

Before presenting the experimental results, we first list in Table 1 some frequently-used acronyms in the following sections.

Table 1. Descriptions of the Frequently-Used Acronyms

View Table

3.1 Spectra

In Fig. 4, we show the representative spectra of three FBG samples with no (on-axis), medium (H-4), and large (V-8) offset distances, characterized under the two orthogonal directions of linear polarization (DoLP-L in blue, DoLP-S in red). In each of the three sub-figures ((a)-(c)), both the transmission (upper) and the relative reflection (lower) spectra in linear scale are presented, and the measured data after normalization is represented by the dotted trace. Particularly, the transmission spectra are normalized to the total coupled-in power $\mathcal {T}_0$ via $\mathcal {T}/\mathcal {T}_0$; the relative reflection spectra are normalized by the background reflection $\mathcal {R}_0$ and the total coupled-in power $\mathcal {T}_0$ via $(\mathcal {R}-\mathcal {R}_0)/\mathcal {T}_0$. $\mathcal {R}_0$ and $\mathcal {T}_0$ are obtained at wavelengths longer than the LP$_{01}$ resonance in each of the corresponding traces. In the reflection spectra, we denote the primary peaks as P1, P2, P3, and P4, if exist, as illustrated in Fig. 4(b.2). The relative intensities of the reflection peaks will be discussed in Section 3.3. We also fit the individual peaks by distinct Gaussian functions [49] exhibited as solid curves. Note that the P3s in Figs. 4(b) and 4(c) are fitted by the sum of two Gaussian functions, this is because P3 in fact consists of two closely-positioned secondary peaks, namely, the self-coupling of the LP$_{11}$ and the cross-coupling of the LP$_{01}$ and LP$_{21}$. However, due to the clear separation between the two secondary peaks in P3, it does not affect our investigation on the self-coupling of the LP$_{11}$. Unlike P3, P1 is formed by the self-coupling of the LP$_{01}$, and P2 corresponds to the cross-coupling of the LP$_{01}$ and LP$_{11}$, which we signify using a virtual mode LP$_{{01}\leftrightarrow {11}}$. Moreover, P4 is likely referring to the cross-coupling of the LP$_{11}$ and LP$_{21}$. The existence of the LP$_{21}$ as revealed by the P3 and P4 is due to the short length of the TM fiber which has a relatively large V-number ($\sim 4.6$). Similar excitation of the LP$_{21}$ has also been observed in UV-inscribed TM-FBGs [50]. Therefore, for the TM-FBGs with moderately long length, the Bragg wavelengths can be expressed as [51]

 

Fig. 4. Representative transmission (upper) and reflection (lower) spectra of (a) on-axis, (b) H-4, and (c) V-8. Traces in blue: DoLP-L; traces in red: DoLP-S; dotted traces: measured data; solid traces: Gaussian-fitted curves.

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3.2 Offset-dependent birefringence

The off-axis position as well as the ellipsoidal shape of the RIM breaks the axial symmetry of the pristine fiber and induces birefringence (${\Delta }n_{B}$). In this section, we show that the induced birefringence is associated with the offset position of the damage points. In theory, the LP$_{11}$ has a 4-fold degeneracy concerning both polarization and spatial distribution, whereas the LP$_{01}$ possesses a 2-fold degeneracy regarding only polarization [2]. We therefore utilize the LP$_{01}$ for our purpose to avoid ambiguity. Here, refer to Fig. 4, we estimate the ${\Delta }n_{B}$ from the Bragg wavelengths of the LP$_{01}$ under the two orthogonal directions of linear polarization (DoLP-L, DoLP-S):

 

Fig. 5. Induced birefringence of all H-$Y$ (green) and V-$X$ (yellow).

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3.3 Polarization- and offset-dependent peak reflectivity

We next study the relative peak reflectivity of all the FBG samples. Since the fiber we used supports more than one mode, generally, selective mode excitation is required to characterize the peak reflectivity of each individual mode (i.e., LP$_{01}$, LP$_{11}$, and LP$_{{01}\leftrightarrow {11}}$). To achieve this, we first fix the working wavelength to the corresponding Bragg wavelength (e.g., $\lambda _{01}^L$), then manually fine-tune the position of the fiber proximal end until the normalized reflected signal $\mathcal {R}$ (see Eq. (1b)) reaches its maximum. Such a $\mathcal {R}_{\textrm {max}}$ after normalization via $(\mathcal {R}_{\textrm {max}}-\mathcal {R}_0)/\mathcal {T}_0$ (the same as in Section 3.1 for the relative reflection spectra) signifies the relative peak reflectivity of the corresponding mode. Note that $\mathcal {R}_0$ and $\mathcal {T}_0$ are obtained under the same excitation condition for $\mathcal {R}_{\textrm {max}}$ of that particular mode. It is also worth noting that due to the unpredictable scattering losses at the resonance [52], the characterization of the absolute peak reflectivity based on the transmission spectra alone may be inaccurate. All the three modes are characterized under the two orthogonal directions of linear polarization (DoLP-L, DoLP-S). The results are presented in Figs. 6(a) and 6(b). The three LP modes are represented in distinct patterns, while the two directions of linear polarization are shown in different colors and are denoted by the superscripts L (DoLP-L) and S (DoLP-S). The additional error bars indicate the measurement error mainly from the manual fine-tuning. For reference, the total loss of the LP$_{{01}\leftrightarrow {11}}^{\textrm {S}}$ of sample V-6 (the highest bar in Figs. 6(a) and 6(b)) is measured as $72.35\pm 0.23\%$ from the transmission spectra alone. From the bar graphs, it is evident that the peak reflectivity depends on both the offset and polarization. In principle, one can relate the peak reflectivity to the overlap between the transverse profile of the grating ($\Delta \varepsilon (x,y)$) and the normal modes of the unperturbed fiber (${\textbf {E}}_k$, ${\textbf {E}}_n$) through the coupling coefficient ($C_{kn}$) derived under the approximation of weak perturbation [41]:

 

Fig. 6. Relative peak reflectivity with error bars of (a) H-$Y$ and (b) V-$X$. Each reflected mode is presented in distinctive pattern; blue bars: DoLP-L; red bars: DoLP-S. (c) Overlap between the damage points (green for H-$y$, yellow for V-$x$) and the ideal fiber modes (LP$_{01}$ and two degenerate spatial LP$_{11}$).

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Apart from the offset dependence, of more interest is the polarization dependence of the relative peak reflectivity. As shown in Figs. 6(a) and 6(b), the peak reflectivity of the LP$_{01}$ is almost constant, whereas the peak reflectivity of the LP$_{11}$ exhibits heavy dependence on polarization, and such dependence differs between the H-$Y$ and V-$X$. The much stronger peak reflectivity of the LP$_{11}$ constantly corresponds to the polarization perpendicular to the damage-point-to-center line. Conventionally in standard SM fibers, the polarization discrimination capability of FS-inscribed FBGs is usually achieved via increasing the eccentricity of the induced RIM [32]. In contrast, FM-FBGs exhibit such strong capabilities (with averaged ratio $7.86$ of the LP$_{11}$ compared to $1.46$ in [32]) even with highly-localized RIM, which we believe stems from the radial asymmetry of the LP$_{11}$. As for the peak reflectivity of the LP$_{{01}\leftrightarrow {11}}$, it has similar polarization dependence to that of the LP$_{11}$ as expected. This is because the reflected LP$_{{01}\leftrightarrow {11}}$ is in fact formed by the cross-coupling of the LP$_{01}$ and LP$_{11}$, the polarization dependence of its peak reflectivity is therefore dominated by that of the LP$_{11}$ since LP$_{01}$ exhibits almost no polarization dependence.

3.4 Polarization-controlled mode patterns

In addition to the peak reflectivity, we find that the reflected mode pattern of the LP$_{{01}\leftrightarrow {11}}$ is also linked to polarization. As is well known, the transverse mode intensity profile of the LP$_{11}$ exhibits certain spatial orientation. Further, under a certain DoLP, theoretically there are two degenerate spatial LP$_{11}$, namely, the LP$_{\textrm {11a}}$ with dumb line parallel to the DoLP and the LP$_{\textrm {11b}}$ with dumb line perpendicular to the DoLP. Here the dumb line is defined as the axis of symmetry that separates the two lobes of the LP$_{11}$. Particularly, in TM fibers with FBG inscribed, the mode profile of the reflected LP$_{{01}\leftrightarrow {11}}$ is resulted from the interference of the LP$_{01}$ and LP$_{11}$ reflected by the grating. The profile orientation, however, can be merely characterized by the LP$_{11}$ due to its radially asymmetric profile. In Fig. 7(a), we show the reflected LP$_{{01}\leftrightarrow {11}}$ fields of sample H-4. In comparison, the reflected LP$_{11}$ fields are also displayed. The images are captured by the camera at the corresponding Bragg wavelengths. For each reflected mode (LP$_{{01}\leftrightarrow {11}}$, LP$_{11}$), three different randomly-chosen excitation conditions (EC1-EC3) under each of the two orthogonal directions of linear polarization (DoLP-L, DoPL-S) are tested. The excitation conditions are altered by tuning the position of the fiber proximal end, and it in fact determines the mode components (e.g., the proportions of the LP$_{01}$, LP$_{\textrm {11a}}$, and LP$_{\textrm {11b}}$) of the optical field right before the grating section. Note that the $\sim 20$-cm-long fiber pigtail between the fiber proximal end and the FBG is kept as straight as possible. Clearly in Fig. 7(a), the orientation of the reflected LP$_{{01}\leftrightarrow {11}}$ fields remains constant under varied excitation conditions under a fixed DoLP. This reveals that the forward LP$_{01}$ couples backwardly with strong preference to one spatial LP$_{11}$. On the contrary, the orientation of the reflected LP$_{11}$ fields varies randomly under different excitation conditions, which indicates that both LP$_{\textrm {11a}}$ and LP$_{\textrm {11b}}$ are excited and involved in the self-coupling of the LP$_{11}$. We next fix the position of the fiber proximal end, and investigate the polarization dependence of the reflected LP$_{{01}\leftrightarrow {11}}$ fields. Presented in Fig. 7(b) are the captured images of V-4 and H-4 at the corresponding Bragg wavelengths under varied DoLP marked by the arrow. As revealed in Fig. 7(b), the orientation of the reflected LP$_{{01}\leftrightarrow {11}}$ fields are determined by the polarization, with the dumb line constantly parallel to the DoLP. Additionally, the wavelength dependence of the reflected LP$_{{01}\leftrightarrow {11}}$ fields is studied as well. We show in Fig. 7(c) the reflection spectrum of the LP$_{{01}\leftrightarrow {11}}$ of sample V-2, along with the associated reflected images at wavelengths noted in white. It is evident that the mode patterns of the reflected LP$_{{01}\leftrightarrow {11}}$ exhibit no wavelength dependence within the reflection peak. It should be noted that although only samples H-4, V-4, and V-2 are exampled in Fig. 7, the properties presented right above are shared by all FBG samples fabricated. In conclusion, for PbP-inscribed TM-FBGs, the forward LP$_{01}$ couples backwardly with strong preference to one spatial LP$_{11}$ with dumb line parallel to the DoLP, suggesting that one can manipulate the reflected mode patterns by merely altering the polarization. Although the orientation preference of the LP$_{11}$ was mentioned in [53], it nevertheless studied the associations of the reflectivity with the orientations of the incident LP$_{11}$, and without polarization control. Moreover, the studies in [34] with respect to the spatial rotation of the LP cladding modes to the incident polarization identified the couplings between the LP$_{01}$ to the two spatial degenerate LP cladding modes, which differs from our findings presented above.

 

Fig. 7. (a) Mode intensity profiles of the reflected LP$_{{01}\leftrightarrow {11}}$ and LP$_{11}$ of sample H-4 at corresponding Bragg wavelengths. (b) Reflected LP$_{{01}\leftrightarrow {11}}$ fields at varied DoLP marked by the double-headed arrow of sample V-4 and H-4. (c) Reflection spectra and mode intensity profiles of the reflected LP$_{{01}\leftrightarrow {11}}$ for sample V-2.

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4. Conclusion

In this work, we experimentally investigate the off-axis FBGs with varied horizontal and vertical distances off the center in a step-index TM fiber fabricated by an FS laser PbP technique. Due to the ellipsoidally shaped damage points induced by the FS laser, the FBG orientation is first characterized through the polar plots of the scattering patterns generated by the visible light launched into the fiber containing the FBG. With the polarization control and visualization of the grating-reflected signals, we characterize the offset dependence of the induced birefringence and of the peak reflectivity. We find that the peak reflectivity of the LP$_{01}$ is almost constant under varied directions of linear polarization, whereas the peak reflectivity of the LP$_{11}$ has strong polarization discrimination capability, with the much stronger reflectivity constantly corresponding to the polarization perpendicular to the damage-point-to-center line. Such polarization discrimination can potentially benefit the fiber laser applications. For example, one can control the lasing lines in [18–20] simply by adjusting the polarization state of the injected light, instead of using an external polarization controller. Moreover, we report that the forward LP$_{01}$ couples backwardly with strong preference to one spatial LP$_{11}$ with dumb line parallel to the direction of linear polarization, through which one can manipulate the spatial orientation of the the radially asymmetric LP modes by merely altering the polarization. With such capability, the transmission capacity of the MDM-based systems mentioned in [1–4] can be further increased via selective mode excitation of arbitrarily oriented radially asymmetric LP modes. Furthermore, the spatial orientation manipulation can also be utilized for cylindrical vector beam [54] and orbital angular momentum [55,56] generations which generally require two spatially orthogonal LP modes with certain phase delays, these can find applications in optical trapping and tweezers [21–23]. Besides, the rather high reflectivities of the cross-coupling of the off-axis FBGs can find immediate applications as mode converters such as in [14,15,57,58], with the bandwidth increased by further chirp designs.