1. Introduction

With the ever-increasing capacity in transmission link of single-mode fiber (SMF), the traditional dense-wavelength-division multiplexing (DWDM)-based fiber link is rapidly approaching the so-called Shannon capacity limit imposed by the Kerr nonlinearity of the SMF itself [1]. Whereas the orbital-angular-momentum (OAM)-based mode division multiplexing (MDM) technique in combination with the yet-established DWDM is considered to be one of the promising solutions to overcome such capacity crunch [2–4]. To make OAM-based MDM technique really available to the fiber communication system, the OAM mode generator/converter, especially the all-fiber one which enables to convert the lower-order mode directly into a specific higher-order OAM mode and vice versa, then becomes the indispensable component. To date, various methods using either the cylindrical lens, the q-plate, the J-plate, the integrated silicon device, or the fiber gratings have been developed to generate the OAM modes [5,6]. Among which, attributed to their unique advantages like the miniature size, low insertion-loss, high conversion-efficiency, and the intrinsic compatibility with the fiber devices, helical long-period fiber grating (HLPG)-based OAM mode generators have recently attracted significant research interest [7–20]. The HLPG-based OAM mode generators written in various fibers like the photonic crystal fibers (PCFs) [9,10], the single mode fibers (SMFs) [11], the few-mode fibers (FMFs) [12–14], the ring-core fiber [15], and the graded-index fiber [16] etc., have already been demonstrated. However, most of the HLPG-based OAM generators mentioned above are limited to either the single-wavelength or the single-band ones [17–20]. The multi-wavelength OAM mode generators, especially the DWDM-channelized all-fiber OAM mode generators have rarely been demonstrated both theoretically and experimentally, such devices can be practically used as an OAM comb filter and thus have potential applications to not only the DWDM-based OAM fiber communications [3,4], but also the DWDM-based OAM comb laser, the OAM holography, and the OAM sensing system well [21–24].

On the other hand, by exploiting the phase-only sampling method [25–27], we have recently proposed and demonstrated multichannel HLPGs [28–31]. In these HLPGs, the optimized phase-only sampling function is equivalently incorporated into the phase of the seeded HLPG either by changing the local pitches of the HLPG [28] or by superimposing a number of the sampling gratings to the seeded HLPG itself, i.e., the so-called DC-sampling method [29–31]. However, due to the inherent characteristic of the large bandwidth (about tens of nanometers) of the seeded HLPG itself in comparison with that of the FBG [25,28], the multichannel HLPGs with a channel spacing larger than 30 nm and few channel-count (up to nine) have only been obtained. To overcome the above issue, most recently, Wu et al. have proposed and demonstrated another multichannel OAM mode generator [32], in which two identical but cascaded long-period fiber gratings (LPGs) with a separation of 46 cm are especially utilized to form an all-fiber M-Z interferometer. As such, a multichannel filter with a channel count up to 17 and a channel spacing of ∼1.7 nm has been successfully demonstrated. However, the proposed device is of the LPGs where the excited higher-order azimuthal mode is inherently not the OAM mode one, thus an additional bulk component, e.g., an in-line polarizer rotator is essentially demanded to obtain the OAM mode [32], which considerably increases the system size and thus decrease the stability and flexibility of the proposed multichannel OAM mode generator. More importantly, the channel spacing obtained there is still too large, which does not match with the DWDM channel grid (i.e., ∼0.8 nm/∼0.4 nm in wavelength unit or 100 GHz/50 GHz in frequency unit). Moreover, the resulted multichannel inherently is of the strong non-uniformity in both the channel amplitude and the channel spacing due to the large group-dispersion difference of the coupled modes in LPGs/HLPGs [33], which inevitably makes the abovementioned device not available to practical applications.

In this study, a full C-band-covered and DWDM channelized OAM mode generator is proposed and experimentally demonstrated, which is realized especially by using a HLPG in combination with a phase-only sampled MFBG. As a proof-of-concept example, the DWDM channelized two complementary 51-channel first-order OAM mode (l = 1) generators have been successfully demonstrated. To the best of our knowledge, this is the first time for proposal and experimental demonstration of such a high channel-count and DWDM channelized OAM mode generator.

2. Configuration and principle scheme of the DWDM-channelized high channel-count all-fiber OAM mode generator

Figure 1 shows the configuration and principle scheme of the proposed DWDM-channelized high channel-count all-fiber OAM mode generator, which consists of a MFBG, a circulator and two identical HLPGs (i.e., HLPG1 and HLPG2). Specifically, there the component MFBG represents a phase-only sampled linearly-chirped FBG, which has two comb-like but complementary spectra in its transmission (T) and reflection (R), like the insets (1) and (2) shown in Fig. 1, respectively. Here it must be noted that the channel spacing among all the channels remains a constant of 0.8/0.4 nm (100 GHz/50 GHz), which is fully consistent with that of the DWDM-channel grids. Therefore, here the MFBG acts like a DWDM-channelized filter but being able to be operated in both the transmission and reflection directions simultaneously. Such kinds of FBGs had been originally proposed and used as FBG-based broadband dispersion compensators [25–27], which can be fabricated by using the advanced phase-shifted phase-mask writing technique [34,35]. Whereas the components HLPG1 and HLPG2 represent the HLPG-based OAM mode converters [12–18], which are specially designed and fabricated for purpose of this study, enabling to turn the fundamental mode HE₁₁ to an OAM mode with 1^st azimuthal order (l = 1) and 10^th radial order efficiently [Appendix]. Both of them have a broad flat-top loss-band in the transmission spectrum as shown in insets (3) and (7) of the Fig. 1, respectively, and the flat-top pass-band in the cross-transmission (CT) spectrum as shown in insets (4) and (8), respectively, where the cross-transmission spectrum refers to the one measured in the fiber cladding region, which is completely complementary to the transmission spectrum. For simplicity, the HLPG1/HLPG2 used in this study are fabricated in a single-mode fiber (SMF). As a result, since this is a SMF, all OAM modes are cladding modes. However, the OAM modes may be guided by the glass-air interface where there is no coating on the fiber, for instance in the region of the HLPG. Once the OAM mode reaches the polymer coated part of the fiber, the light is absorbed. Such kinds of HLPGs have already been reported and demonstrated by using the dispersion-turning-point (DTP)-based HLPG [17–20] and the phase-shifted HLPG as well [36].

 

Fig. 1. Principle and configuration scheme for the proposed DWDM-channelized all-fiber OAM mode generator, where the MFBG represents phase-only sampled linearly-chirped FBG. HLPG1 and HLPG2 represent the HLPG-based OAM mode converters. The insets (1) and (2) represent the transmission and reflection spectra of the utilized MFBG, respectively. The insets (3)/(7) and insets (4)/(8) represent transmission and cross-transmission spectra of the utilized HLPG1 and HLPG2, respectively. The insets (5) and (6) represent the transmission and the cross-transmission of the MFBG + HLPG1, respectively. The insets (9) and (10) represent the transmission and the cross-transmission of the MFBG + Circulator + HLPG2, respectively. Inset (11) represents the spectral summation of the inset (6) and (10), which indicates that two complementary sets of multi-channels can be obtained.

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To make full use of the MFBG, two of its operation states, i.e., transmission and the reflection, are both considered in this study. When the MFBG is operated in its transmission direction (as shown on right side of the Fig. 1), only the MFBG and HLPG1 are required. As such, the transmission and the cross-transmission spectra, like the ones shown in insets (5) and (6) can be obtained. Particularly, the result shown in inset (6) implicitly means that the DWDM channelized multichannel OAM mode (l = 1) generation can be realized after the HLPG1. On the other hand, when the MFBG is operated in its reflection direction (as shown on left side of the Fig. 1), besides the MFBG and the HLPG2, the circulator is additionally demanded. Transmission and cross-transmission spectra like the ones shown in insets (9) and (10) then can be obtained. Similarly, the spectrum shown in inset (10) means that the DWDM channelized multi-channel OAM mode (l = 1) generation can also be realized after the HLPG2. More importantly, it is worth noting that, as shown in inset (11), the obtained OAM multichannel spectra shown in the insets (6) and (10) are the mutual complementary ones (attributed to the complementarily spectral characteristics of the MFBG in transmission and reflection). The above result implicitly means that two complementary multichannel OAM mode generator can be obtained. In other words, the channel-count of the proposed multichannel OAM mode generator can be further doubled just by making the MFBG work in both its transmission and reflection simultaneously.

3. Measurement results for the utilized MFBG and the broadband HLPG

The MFBG practically used in this study is a multichannel FBG with a channel-count up to 51, which was optimally designed by using the phase-only sampling method and fabricated by using the phase-shifted phase-mask writing technique [26]. This kind of FBG was originally developed and utilized as a FBG-based dispersion compensator [35]. Meanwhile, two identical HLPGs, i.e., HLPG1 and HLPG2 utilized in this study were optimally designed by using the DTP method [Appendix] and fabricated thereafter by using the sapphire tube-based CO₂ laser direct-writing technique [37].

Figure 2 shows the measurement results for reflection and transmission spectra of the utilized MFBG, and transmission spectrum of the utilized HLPG, respectively, where Figs. 2(a) and 2(b) show the whole 51-channel spectra of MFBG in the transmission and reflection, respectively. More specifically, Figs. 2(a1)–(a3) show transmission spectra of the enlarged 3 channels at the leftmost, middle, and rightmost of the 51 channels, respectively. Whereas the Figs. 2(b1)–(b3) show the reflection spectra of the enlarged three channels at the leftmost, middle, and rightmost of the 51 channels, respectively. From these figures, it can be seen that a full C-band (ranging from 1525.8 to1565.8 nm) covered 51-channel FBG with excellent inter- and intra-channel uniformities in both the transmission and the reflection spectra (including the channel amplitude, the channel bandwidth and the channel spacing) has been successfully obtained. The grating strength is larger than 10 dB (i.e., the reflection is larger than 90%). The channel spacing among all the channels remains a constant of 0.8 nm and the intra-channel flat-top bandwidths in transmission and reflection are ∼0.45 nm, and ∼0.25 nm, respectively. In addition, to compare the spectra shown in Figs. 2(a1)–(a3) with those shown in Figs. 2(b1)–(b3), respectively, one can find that the 51-channels generated in transmission are complementary to the 51-channels generated in reflection, which is completely the same as what we expect in Fig. 1, in return indicates that the utilized phase-only sampled FBG could be potentially used as an optical wavelength interleaver.

 

Fig. 2. Measurement results for transmission and reflection spectra of the phase-only sampled 51-channel FBG, and transmission spectrum of the utilized HLPG. (a)Transmission spectrum covering the whole 51 channels. (b) Reflection spectrum covering the whole 51 channels. The zoomed 3 channels in the transmission spectrum are the leftmost (a1), middle (a2), and rightmost (a3) of the whole 51 channels. The zoomed 3 channels in the reflection spectrum are the leftmost (a1), middle (a2), and rightmost (a3) of the whole 51 channels. (c) Transmission spectrum of the utilized HLPG.

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For clarity, the measurement result for the utilized HLPG is also shown, where Fig. 2(c) shows its transmission spectrum. From this figure, it can be seen that a broad band-rejection filter with a bandwidth of ∼40 nm@-10 dB and a central wavelength of ∼1550.0 nm has been successfully obtained. However, the ideal flat-top spectrum like the one shown in Fig. 7 of the appendix cannot be obtained, instead there exists a slope in the envelope of the spectrum at the wavelength ranging from 1525.8 to1565.8 nm, and such non-ideal spectrum could be ascribed by the fact that there inevitably exist some deviations in local pitches of the HLPG during its fabrication. Moreover, the insertion loss of the HLPG measured from the baseline of the spectrum is about 1.5 dB, which is also ascribed by the distortions in the HLPG produced during the fabrication process. Except for the above discrepancies, the experimental results agree well with the simulation ones, which in return indicates that the HLPG could be exploited as a wideband OAM mode converter in this study.

4. Experimental setup and measurement results for the two complementary 51-channel DWDM-channelized all-fiber OAM mode generators

In the following, the proof-of-concept test is performed. Figure 3 shows the experimental setups and the measurement results, where Fig. 3(a) and Fig. 3(b) show the setup and the corresponding results while the MFBG is operated in its reflection and transmission, respectively. In both cases, the same MFBG and HLPG, which have the spectra like the ones shown in Fig. 2, were utilized. In Fig. 3(a), the devices ASE and OSA represent a wide-band amplified spontaneous emission light source and an optical spectral analyzer, respectively. The light emitted from the ASE source propagates according to the route: ASE source-circulator-MFBG-circulator-HLPG2. The inset in Fig. 3(a1) is a measurement of the residual light in the fundamental mode, after the coated part of SMF between the HLPG and the OSA, all light in the higher-order mode has been stripped away. Whereas the inset Fig. 3(a2) shows the spectrum of the OAM mode (i.e., the higher-order cladding mode), which actually is deduced from all the data shown in the inset Fig. 3(a1), since the transmission and cross-transmission spectra of the HLPG2 are complementary each other.

 

Fig. 3. Experimental setups and the measurement results for the cases that the MFBG is utilized in its (a) transmission and (b) reflection, respectively, where the insets (a1) and (a2) show the measurement result and the deduced result for the transmission and cross-transmission spectra after the HLPG2. Whereas the insets (b1) and (b2) show the measurement result and the deduced result for the transmission and the cross-transmission spectra after the HLPG1, respectively.

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From this figure, it can be obviously seen that a 51-channel comb filter with spectral performances almost the same as the transmission spectrum of the MFBG (i.e., the spectrum shown in Fig. 2(b)) has been successfully obtained. In addition, to compare the spectrum shown in Fig. 3 (a1) with the theoretical one shown in inset (9) of the same Fig. 3(a), it can be found that, except for a small slope (non-flatness) that existed in the envelope of the 51-channel spectrum, which, however, can be ascribed by the fact that the real transmission spectrum of the HLPG2 (i.e., the one shown in Fig. 2(c)) was no longer the flat-top one like the inset (7), the experimental result shown in Fig. 3(a1) agrees well with the theoretical one. The above result in return means that a 51-channel OAM comb like the one shown in the inset (10) of Fig. 1 can be obtained immediately after the HLPG2.

While for the transmission case as shown in Fig. 3(b), the light emitted from the ASE source would pass the MFBG and HLPG directly with few insertion-losses. Similarly, the inset (b1) is also a measurement of the residual light in the fundamental mode, where after the coated part of SMF between the HLPG and OSA, all light in the higher order mode has been stripped away. The inset (b2) represents the spectrum of the OAM mode (i.e., the higher-order cladding mode), which is indirectly deduced from the data shown in inset (b1). From this figure, it can be seen a 51-channel comb filter with a spectrum exactly complementary to the one shown in the inset (a2) of Fig. 3(a) has been successfully obtained. To compare the spectrum shown in inset (b1) with the theoretical one shown in inset (5) of Fig. 1, it can be found that, except for the small slope (non-flatness) existed in the envelope of the 51-channel spectrum, which can be ascribed by the fact that the real transmission spectrum of the HLPG1 is no longer the ideal flat-top one like the inset (3) of Fig. 1. The experimental result agrees well with the theoretical one, which in return indicates that a 51-channel OAM comb like the one shown in the inset (6) of Fig. 1 can really be obtained after HLPG1.

5. OAM channel performances for the proposed high channel-count multichannel OAM mode generator

To further reveal that all the 51-channels shown in the insets (a2) and (b2) of Fig. 3, i.e., the 51-channel cross-transmission spectra, are of the transmitted light in OAM modes, the intensity and phase distributions of the converted OAM mode were measured immediately after the HLPG1/HLPG2. Particularly, the measurement setup for the transmission case (i.e., the case of Fig. 3(b)) is shown in Fig. 4, which is almost the same as the ones that we had utilized in Refs. [12–14], where a single-longitudinal-mode and wavelength-tunable laser with a wavelength ranging from 1490 to 1590 nm and a wavelength resolution of 1 pm is used as the light source. The principle used to detect the phase-distribution of the OAM mode is based on the investigation of the interference pattern produced by combining a laser beam emitted at the output end of the utilized HLPG with the reference beam tapped from the same laser source. Moreover, polarization status of the light source has not been considered in this study. Details about all the components in this setup could be found in our previous papers [12–14].

 

Fig. 4. Experimental setup for measuring the OAM performances of the proposed system.

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Fig. 5. Intensity and phase distributions of the first-order OAM modes measured at central wavelengths of the three typical channels: 1st, 26th, 51st channels, respectively. The results were obtained in the cases when the MFBG was operated in its (a) reflection and (b) transmission, respectively.

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Figure 5 shows the intensity and the spiral petal-like interference patterns measured at the central wavelengths of three typical channels, i.e., the left-most (namely the Ch.1), the central (namely the Ch.26) and the right-most channels (namely the Ch.51) of the two 51-channel comb filters. For comparison, the corresponding channel spectra are also depicted in Fig. 5, where Fig. 5(a) shows the results obtained when the MFBG is operated in the reflection direction. The two insets shown in the spectra of the three channels (Ch.1, Ch.26, and Ch.51) represent the intensity and phase distributions of the output mode, respectively. From this figure, it can be seen that the first-order OAM modes but with a radial index of 10 have been successfully generated at the wavelengths of 1526.25 nm, 1545.93 nm, and 1566.11 nm, respectively, and the OAM mode conversion efficiency is estimated to be 90% supposing that the excited higher-order azimuthal modes are of the purely OAM one. Here it must be noted that the intensity distributions shown in these figures are obtained while the reference beam is blocked and the 40x lens is replaced by a 10x one in the setup as shown in Fig. 4, so that the whole area of the fiber cladding can be clearly imaged, but at cost, the fiber core and the most inner ring cannot be discerned from these images. The result to obtain the OAM modes with a radial index of 10 agrees well with what we expect in the Appendix. Whereas Fig. 5(b) shows the results obtained while the MFBG is operated in the transmission direction. From this figure, it can be seen that the first-order OAM modes at the wavelengths of 1526.65 nm, 1546.33 nm, and 1566.51 nm have also been generated.

Here it should be noted that for simplicity, the OAM modes generated at the three typical channels, namely Ch.1, Ch.26 and Ch.51, were demonstrated only in Fig. 5, as matter of fact, observations for the newly-generated OAM mode through all the 51 channels have been accomplished and nearly the same ones as those shown in Fig. 5 can be obtained. Furthermore, to compare the channel spectra shown in Fig. 5(a) with those shown in Fig. 5(b), it can be obviously seen that these two 51-channel spectra are fully complementary to each other, which means that the proposal to double the channel-count of the MFBG + HLPG by further interleaving such two 51-channel comb filters is fairly correct. All the above results fully verify that the proposed MFBG incorporated with HLPGs can really be used as a DWDM channelized high channel-count OAM mode generator.

Here it must be noted that since the HLPGs utilized in this study were fabricated in a conventional SMF, the OAM mode (l = 1) is generated in the cladding region of the fiber, which in fact cannot be practically used in an optical fiber system due to the considerably high propagation loss in such region. However, such issue can easily be overcome if the few-mode fibers (FMFs) or the other kinds of multimode fibers (MMFs) are utilized to write the HLPGs, because the higher-order, e.g., the second- and the third-order OAM mode converters have already been reported and demonstrated in [12–14]. With such techniques, the higher-order OAM modes generated in the core region of either the FMFs or the MMFs can then be obtained. Moreover, unlike the conventional HLPG-based OAM mode generator, the proposed multichannel OAM mode generator can be operated simultaneously in two states, namely the transmission and the reflection directions of the MFBG, which provides a novel and simple method enabling to produce the spectrally-complementary two multichannel OAM mode generators with high conversion efficiency and yet with great flexibility in controlling the order number of the two sets of the OAM modes independently. In other words, it is easy for us to combine two different kinds of OAM channels, e.g., if the HLPG1 is designed as the first-order OAM mode converter meanwhile the HLPG2 is designed as the second-order OAM one, then the combination of high channel-count first- and second-order OAM mode generator can be realized.

6. Conclusion

In this study, a full C-band covered and DWDM channelized OAM mode generator is firstly proposed and experimentally demonstrated, which is realized by using a phase-only sampled MFBG and two HLPGs. The utilized MFBG is of the phase-only sampled FBG with the in-band channel number of 51, which functions like a DWDM-channelized filter capable of operating in both the transmission and reflections. Whereas the two HLPGs function as broadband OAM mode converters, which enables to convert the fundamental mode HE₁₁ into the demanded OAM mode efficiently at wavelengths through the full C-band. As a proof-of-concept example, the DWDM channelized two complementary 51-channel OAM mode generators have been successfully demonstrated, each of which has a channel spacing of 100 GHz (∼0.8 nm), a useful bandwidth of ∼40 nm, and high uniformities in both inter- and intra-channel spectra as well. To the best of our knowledge, this is the first time for proposal and experimental demonstration of such high channel-count and DWDM channelized OAM mode (l = 1) generator, which could also be used to produce multichannel higher-order OAM mode generators as long as the utilized HLPGs are capable of generating the broadband higher-order OAM modes. The proposed device has potential applications to the DWDM-based OAM fiber communications, OAM comb lasers, OAM holography, and OAM sensors as well.

Appendix

Design and fabrication of the broadband flat-top helical long-period fiber grating (HLPG)

Helical long-period fiber grating (HLPG), sometimes also called chiral fiber grating, is one kind of long-period fiber grating (LPG) that refers to a piece of fiber where there exists a periodical screw-type index-modulation along the fiber axis [7,8]. Attributed to the unprecedented properties like the intrinsic orbital-angular-momentum (OAM) modes, compact size, and the inherent compatibility with the conventional fibers etc., HLPGs have attracted special research interest and found versatile applications to OAM mode converters [9–15]. Here we have proposed and demonstrated an all-fiber broadband OAM mode converter, which is based on a HLPG but operated precisely at the dispersion-turning-point (DTP) of the mode LP_1,10. Unlike most of the HLPG-based OAM mode converters where the flat-top and the broad-band performances have rarely been obtained simultaneously, here a first-order OAM mode converter with a flat-top bandwidth of ∼49 nm@-10 dB and a conversion efficiency of ∼90% has been successfully demonstrated.

It is generally known that, for LPGs as well as the HLPGs, since the mode couplings occur between the core mode and the distinct cladding modes, the modal dispersion and its effects cannot be neglected, which could strongly affect the spectral performances of the grating, especially to the bandwidth of the rejection filter (notch). For all cases considering the mode dispersion effect, the rejection bandwidth for a HLPG can be approximately expressed as [38]

(1)$$\varDelta {\lambda _B} \approx \frac{{\varDelta {\lambda _0}}}{{|{1 - \varLambda (\lambda )\cdot ({{{d\varDelta {n_{eff}}} / {d\lambda }}} )} |}} = \frac{{\varDelta {\lambda _0}}}{{\varDelta {n_{eff}} \cdot ({|{{{d\varLambda } / {d\lambda }}} |} )}}, $$

where $\Delta {\lambda _B}$ and $\Delta {\lambda _0}$ represent the notch bandwidth with and without considering the mode dispersion effect, respectively. λ represents the wavelength. $\varDelta {n_{eff}}$ represents the effective index difference between the core and the coupled cladding mode. $\varLambda (\lambda )$ represents the HLPG’s pitch. Under the assumptions that the considered HLPG is rather strong with a rejection depth larger than 20 dB ($\kappa L \approx \pi /2$) and the power of the core mode is fully coupled into one particular cladding mode, the 20 dB local resonance peak of bandwidth $\varDelta {\lambda _B}$ can be expressed as [16,39]

(2)$$\Delta {\lambda _B} \approx 0.06\kappa \frac{{{\Lambda _0}^2}}{{|{{{\textrm{d}\Lambda } / {\textrm{d}\lambda }}} |}},$$

where $\kappa $ represents the coupling coefficient between the mode LP_0,1 and the coupled higher-order mode. From the Eq. (2), it is easy to find that bandwidth Δλ_B is inversely proportional to |dΛ/dλ|. In other word, the broadest bandwidth can be achieved if the HLPG is operated at a nominal wavelength λ_0, namely the dispersion-turning-point (DTP) wavelength, where the condition dΛ/dλ=0 is satisfied. In such case, bandwidth Δλ_B can no longer be determined by Eq. (2), instead, it can be determined by [40]

(3)$$\varDelta {\lambda _B} \approx {\varLambda _0}{\left\{ {\frac{{0.06\kappa }}{{{{|{{{{d^2}\varLambda } / {d{\lambda^2}}}} |}_{\lambda = {\lambda_0}}}}}} \right\}^{_{1/2}}}. $$

Equation (3) obviously indicates that, for the LPG as well as the HLPG, the bandwidth $\Delta {\lambda _B}$ is inversely proportional to the magnitude of ${{d\varLambda } / {d\lambda }}$. Therefore, it is easy for one to come across the conclusion that if the HLPG is arranged to operate at or very close to the DTP wavelength, the obtained rejection spectrum would be of the flat-top and the broad one, in other word, the broad and the flat-top bandwidth  Δλ_B can be obtained simultaneously.

In order to validate the above presumption, we optimally designed the HLPGs, which are assumed to be fabricated in conventional single-mode fiber. All the parameters, such as the diameters of the core and the cladding are assumed to be 125 µm and 8.2 µm, respectively. The refractive indices of the core, the cladding, and the surrounding material are assumed to be, 1.458, 1.4532, and 1.0, respectively. Moreover, the wavelengths ranging from 1400 nm to 1700nm are particularly considered in this study. By numerically solving the dispersion equations described in [41], we obtained the pitch spectra for different cladding modes, which are shown in Fig. 6(a), where the curves LP_1,8, LP_1,9, …, and LP_1,12 represents the cases that the resonant couplings occur between the mode LP_0,1 and the modes LP_1,8 -LP_1,12, respectively. It can be seen that under current fiber parameter conditions, within the C-L bands of the fiber communication, i.e., the wavelengths ranging from 1500 nm to 1600 nm, there exists a DTP only for the mode of LP_1,10. To observe the DTP pitch more clearly, the curve LP_1,10 shown in Fig. 6(a) is enlarged and individually depicted in Fig. 6(b). From Fig. 6(b), it can be seen that at the point of (1560 nm, 222.32 µm), the magnitude of ${{d\varLambda } / {d\lambda }}$ is precisely equal to zero. The HLPG with a rejection depth of 10 dB was considered in the simulations, which is particularly designed to have a pitch exactly as 222.32 µm. In accordance, the mode coupling occurring between the LP_0,1 and LP_1,10 at wavelengths ranging from 1400 to 1700nm is considered in this study. Transmission spectrum of such HLPG is calculated and the result is shown Fig. 7, where the index-modulation is assumed to be 1.64 × 10⁻⁴. The length of the HLPG is assumed to be 4.0 cm (with 180 periods). From Fig. 7, it is seen that while the grating’s pitch is assumed to be 222.32 µm, a band-rejection filter with a flat-top bandwidth of ∼49 nm@-10 dB has been successfully obtained, which is located at a central wavelength of ∼1550 nm. The inset represents the normalized intensity distribution of the LP_1,10 mode at the wavelength of 1550 nm.

 

Fig. 6. (a) The pitch spectra of the LPG for modes LP_1,8-LP_1,12. (b) The pitch spectrum for mode LP_1,10.

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Fig. 7. Simulation result for transmission spectrum of the designed HLPG where the maximum index-modulation is assumed to be 1.64 × 10⁻⁴ and the pitch is assumed to be 222.32 µm. The inset shows the normalized intensity distribution of the LP_1,10 mode at the wavelength of 1550 nm.

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To experimentally verify the above proposal, the designed HLPG was fabricated by using the SMF. The grating pitch was especially adopted as 222.32 µm. The fabrication setup is the same as what we had used in [37]. Unlike the other CO₂ laser direct-writing techniques which in general the fiber is directly heated by the laser beam through a focused lens, here a sapphire tube is specially designed and utilized in place of the focused lens, as a result, the fiber within the tube region can be homogeneously heated and twisted. More importantly, the pitch of the HLPG can be precisely controlled by adjusting the moving speed of the fiber-loaded stage and the rotator speed, accuracy of the pitch obtained in our setup is estimated to be ±0.5 µm, which is mainly determined by position precision of the utilized translation-stage.

Funding

Yazaki Memorial Foundation for Science and Technology (2020-08); Japan Society for the Promotion of Science (JP 22H01546); Natural Science Foundation of Jiangsu Province (BK20201370).

Acknowledgments

We thank TeraXion Inc. for providing the 51-channel FBG samples. We would also like to thank the anonymous reviewers for their valuable and constructive comments and suggestions.

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 authors upon reasonable request.

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