Optimal design and fabrication of multichannel helical long-period fiber gratings based on phase-only sampling method

Chengliang Zhu; Shoma Ishikami; Peng Wang; Hua Zhao; Hongpu Li

doi:10.1364/OE.27.002281

1. Introduction

In the past few decades, long-period fiber grating (LPG) have been comprehensively studied and have been found versatile applications in the fields of optical communications, all-optical signal processing, and optical sensing for civil engineering [1–21]. To date, various methods including the ultraviolet (UV) exposure [1,2], the electric-arc discharge [22,23], the mechanical pressure [24,25], and the focused CO₂ laser radiation [26] have been developed to fabricate LPGs. Among all the previous ones, the gratings fabricated by using a CO₂ laser-based point-by-point writing technique [27] have attracted much more attentions due to the inherent advantages, e. g., extreme low-cost in terms of both writing and processing devices; high flexibilities for the fiber’s selection (almost suitable for any kinds of the fibers), robust performances, and ultrahigh temperature stability (can work at a temperature higher than 800 °C). In terms of the different applications, various kinds of LPGs have been developed to date, which includes the conventional LPG, the phase-shifted LPG, the cascaded LPGs with nanostructured coatings, the helical-type long-period fiber gratings, etc [28–35]. Attributed to the intrinsic helicity characteristics which are especially suitable for controlling the polarization and orbit-angular-momentum (OAM) state of the light in optical fiber, the helical-type LPG (HLPG) has recently attracted a great research interest and has been found versatile applications, such as the temperature and torque sensors, all-fiber band-rejection filter, mode-converter for micromanipulation, and conversion of orbital angular momentum beams, etc. [34–45]. Most recently, based on the utilization of CO₂ laser, we have proposed and developed a novel technique to fabricate HLPG [46], where a sapphire tube is specially designed and utilized instead of the commonly used focal lens. As a result, the HLPGs can be robustly fabricated without any defects on the grating’s surface. However, most of the LPGs proposed and demonstrated to date are mainly limited to the single-channel one where there exists only one spectral notch for each of the coupled cladding mode, multichannel LPG, especially multichannel HLPG has never been proposed and demonstrated, which however, would be very useful and essential to wavelength division multiplexing (WDM) fiber sensors [47] and the multi-wavelength OAM mode converters in the field of optical communications [45,48].

On the other hand, fiber Bragg grating (FBG) based multi-channel filters have been systematically studied in the past decade, owing to their unique characteristics for DWDM wavelength filtering [49–60], such as the compact size, low insertion loss, high reliability, and inherent connectivity to any other kinds of fiber devices. To date, various FBG-based multi-channel filter, for example, the amplitude-only sampled FBG, the multiple-phase-shifted FBG, the superimposed FBGs, and the phase-only sampled FBG have been proposed and experimentally demonstrated [49–60], among which the phase-only sampled ones proposed and developed by Li (one of our authors) et al. [57–60] had attracted much more attention because it can reduce the index-change required for a 45- or 81-channel sampled FBG to a practical level. Furthermore, the phase-only sampling has no change in its amplitude such that the index-change profile for a multichannel FBG is exactly the same as that of the seed grating, thus making the multichannel gratings particularly suitable to be fabricated with the robust side-writing phase-mask technique [58]. Until now, the sampling method is an established technique that is generally utilized to produce multi-channel FBG. However, the sampled LPG, especially for the phase-only sampled HLPG has not been proposed and demonstrated yet, except for the one reported and demonstrated by Dong et al. [61] where a rectangular type amplitude-sampling was employed and few channels with a large non-uniformity in both the depth and bandwidth were obtained based on the UV writing technique. Moreover, due the inevitable existence of blank region in each sampling period of the grating, a longer grating is demanded in order to retain the same rejection depth of the LPG.

In this study, a new kind of multichannel HLPG based on the phase-only sampling method has been firstly proposed and experimentally demonstrated, to the best of our knowledge. The proposed method requires the minimum as well as a uniform index-modulation to the designed multichannel gratings, which considerably facilitates the fabrication process by using the CO₂ laser writing technique and makes the multichannel HLPGs to be fabricated in even a conventional single-mode fiber (SMF).

2. Principle of the sampled HLPG and the designed results

Like a sampled FBG, the sampled HLPG is a HLPG whose amplitude is axially modulated by a periodic sampling function S(z) and the index change $Δ n_{M} (z, ϕ)$ can be expressed as [41,57]

Δ n_{M} (z, ϕ) = Re {(Δ n (z, ϕ) / 2 \cdot \exp (i l ϕ)) \exp {i l 2 π z / Λ_{0}}} \times S (z),

where Re represents the real part,

Δ n (z, ϕ)

is the maximum index-modulation, z is the position along the grating, φ represents the azimuthal angle,

Λ_{0}

is the grating pitch, l represents the helicity of the HLPG, and

S (z)

is a sampling function with a period P, which can be expanded in the Fourier series as

S (z) = \sum_{m = - \infty}^{\infty} S_{m} e x p (i 2 m π z / P),

where

S_{m}

is the mth complex coefficient of the Fourier series. Noted that for convenience, the dc part of the index-modulation is neglected in Eq. (1). According to the analyzed results given in the Appendix, it is known that if the strength of the considered LPG is weak enough, the cross-transmission spectrum

t_{c} (β)

of the LPG can simply be obtained by doing the Fourier transform in z direction on both sides of the Eq. (1) as

t_{c} (β) = ℑ (Δ n_{M} (z)) = ℑ (Δ n_{s} (z)) \otimes ℑ (S (z)) = ℑ (Δ n_{s} (z)) \otimes \sum_{m = - \infty}^{m = \infty} s_{m} \cdot δ (β - m \frac{π}{P}),

where

β = (β_{c o r e} - β_{c l a d} - l \cdot {2 π / Λ}_{0}) / 2

represents the grating’s detuning in the wave-number domain,

β_{c o r e}

and

β_{c l a d}

represent the propagation constants for the forward fundamental core-mode and the particular cladding-mode, respectively. The operator

ℑ

represents the Fourier transform,

\otimes

represents the convolution operation, and δ represents the delta function. From the above equation, it is obviously seen that the cross-spectrum of the sampled HLPG become the one, which is a convolution between the spectrum of the seed HLPG and the Fourier transform of the periodical sampling function (which is a delta-series but modulated with the Fourier coefficients). As a result, multiple channel in spectrum with channel spacing of

π / p

(measured in wavelength number domain) will be created and except for the amplitude, the spectral profile of each channel must be identical to that of seed grating. Moreover, since the designed HLPGs will be fabricated by using the conventional single-mode fiber, only the single-helix HLPGs (l = 1) are considered in this study and the channel spacing can be expressed by

Δ λ = \frac{λ^{2}}{{(n_{e f f}^{c o r e} - n_{e f f}^{c l a d}) - λ \frac{\partial (n_{e f f}^{c o r e} - n_{e f f}^{c l a d})}{\partial λ}} \cdot P} \approx \frac{λ \cdot Λ_{0}}{P} (\frac{1}{1 - Λ_{0} \cdot \frac{\partial (n_{e f f}^{c o r e} - n_{e f f}^{c l a d})}{\partial λ}}),

where

n_{e f f}^{c o r e}

and

n_{e f f}^{c l a d}

are the effective index of the fundamental core mode and the particular cladding mode, respectively, λ is the designed nominal wavelength of the HLPG. Noted that in Eq. (4), the dispersion effects for both the core mode and the cladding mode are included.

For the design of a multichannel HLPG, the phase-only sampling method is utilized, which doesn’t require an additional amplitude apodization in each sampling period and thus make the designed HLPG much easier to be fabricated. Without lose any generalities, here phase of the sampling function $φ (z)$ is assumed to have the form including a certain number of harmonic terms expressed by [57,58]:

s (z) = \exp (i φ (z)) = \exp {i \sum_{n = 1}^{J} α_{n} \cos (2 π n z / P)} = \sum_{m = - \infty}^{\infty} S_{m} \exp (i 2 m π z / P),

where the parameters

α_{n}

are optimally selected such that each of the Fourier coefficients

S_{m}

are identical within the considered channel numbers. Based on the symmetric characteristics of the Fourier series

S_{m}

in Eq. (5), it is expected to achieve at most 2J-1 uniform channels once if the number of harmonic terms adopted is assumed to be J. In order to obtain optimal magnitudes of the series of

α_{n}

in Eq. (5), we employed the simulated annealing algorithm [62], which is a recursive metropolis algorithm to reduce a cost function with multiple parameters. The optimization criteria are the uniformities of the channel spacing and the channel amplitudes, and the maximum in-band diffraction efficiency η. Therefore, the cost function can be defined as

E (z) = \sum_{m = - M}^{M} {[{| S_{m} (α_{1}, α_{2}, \dots α_{J}) |}^{2} - \frac{η}{2 M + 1}]}^{2},

where η is the target diffraction efficiency for all the in-band channel (2M + 1), and S_m are the Fourier series of the sampling function

S (z)

. Their summation is referred as the diffraction efficiency

η = {\sum_{m = - M}^{M} | S_{m} |}^{2} .

To confirm the above theory, designs for the phase-only sampling function were completed. Particularly, the optimal results for a 3- and a 9-channels phase-only sampling functions were obtained as shown in Fig. 1 and Fig. 2, respectively. Figure 1(a) shows the phase distribution for the 3-channel sampling function which is normalized in one period and can be analytically expressed by

φ_{3} (z) = 1 .437 \times \cos (2 π z / P) .

Figure 1(b) shows the channel spectrum. From this figure, it can be seen that the non-uniformity for the first three Fourier coefficients,

| S_{- 1} |

,

| S_{0} |

, and

| S_{+ 1} |

is less than 0.5%. The in-band energy (diffraction) efficiency

η = \sum_{m = - 1}^{1} {| S_{m} |}^{2}

is larger than 91%. Figure 2(a) shows the phase distribution of the 9-channel sampling function, which can be expressed in a formula by

\begin{matrix} φ_{9} (z) = -2 .935 \times \cos (2 π z / P)-0 .758 \times \cos (4 π z/ P) \\ + 0 .420 \times \cos (6 π z/ P)-0 .318 \times \cos (8 π z/ P) + 0 .171 \times \cos (10 π z/ P) . \end{matrix}

Figure 2(b) shows the obtained channel spectrum in which the non-uniformity through all the 9 channels is less than 0.4%, and the in-band energy (diffraction) efficiency

η = \sum_{m = - 4}^{4} {| S_{m} |}^{2}

is larger than 97%.

Fig. 1 Optimization result for 3-channel phase-only sampling function. (a) Phase distribution in one period and (b) the corresponding channel spectrum.

Download Full Size | PDF

Fig. 2 Optimization result for 9-channel phase-only sampling function. (a) Phase distribution in one period and (b) the corresponding channel spectrum.

Download Full Size | PDF

To further verify the principle of the sampled HLPG, the optimized sampling functions have been utilized to multiply the seed HLPG in spatial domain. The central pitch Λ₀ is assumed to be 570 μm. Other parameters such as the radii of the core a₁ and cladding a₂, the refractive indices of the core n₁ and cladding n₂, and the surrounding material n₃, are particularly chosen as: a₁ = 4.1 μm, a₂ = 62.5 μm, n₁ = 1.4580, n₂ = 1.4536, and n₃ = 1.0. As a matter of fact, in the transmission spectrum of HLPG, there should exists a series of attenuation band (notch) caused by the coupling between the fundamental mode and the different cladding modes, but in our case only the one corresponding to the LP₁₄ mode is within the wavelength of 1500-1650 nm while the grating period of 570 μm is adopted. Therefore, for convenience, the resonant wavelength of the HLPG resulted from the coupling between the fundamental core mode and the LP₁₄ cladding mode is considered and assumed to be 1560 nm in this study. Transmission spectra of the sampled HLPGs are then calculated by using the transfer matrix method [63,64]. Here it must be pointed that in order to precisely estimate the channel spacing of the resulted multichannel HLPG, dispersion effects on both of the core mode LP₀₁ and the cladding mode LP₁₄ must be considered. Figure 3 shows the calculated results for the transmission spectra of the sampled HLPG, where Fig. 3(a) shows the 3-channel HLPG and Fig. 3(b) shows the 9-channel one. Noted that for both the 3-channel and 9-channel HLPGs, the sampling period is assumed to be 1.710 cm and the total length of the grating is assumed to be about 5.130 cm (with three sampling periods). The maximum index-modulations for the 3-channel and 9-channel HLPGs are assumed to be 2.24x10⁻⁴ and 3.99x10⁻⁴, respectively. From the Fig. 3(a), it can be seen that a 3-channel HLPG with an identical notch depth of 15 dB has been successfully obtained. The channel spacing is 30 nm, which is dependent on the grating pitch, the sampling period of the P, as well as the dispersion effect of the effective index difference between the core mode and the LP₁₄ cladding mode as shown in Eq. (4). Moreover, it can be seen that the side lopes on both sides of the channel are considerably restrained. Whereas from the Fig. 3(b), it can be seen that a 9-channel notch filter with almost an identical channel spacing of 30 nm and almost an identical notch depth of 20 dB has also been obtained. The above results mean that the proposed sampling method to produce multichannel is also available to HLPG and works pretty well even for a strong HLPG with a notch depth up to 20 dB. However, when the HLPG strength is further increased, the Fourier transform relationship between the cross-transmission spectrum $t_{c} (β)$ and the distribution of the index-change may not be precisely satisfied, the obtained channel spectra may be strongly degraded.

Fig. 3 Design results for phase-only sampled HLPGs with (a) 3-channel and (b) 9-channel.

Download Full Size | PDF

3. Fabrication of the phase-only sampled HLPG and the measurement results

To confirm the above design results, a couple of 3-channel HLPGs and 9-channel HLPGs were fabricated. The experimental setup is shown in Fig. 4, which is the same as the one what we had developed and utilized in [43–45] where a single-channel HLPG had been fabricated. Unlike the previous CO₂ writing methods used in [35–38] where a ZnSe lens was generally utilized in order to focus the CO₂ laser beam directly onto the fiber, a sapphire tube is specially designed and utilized in place of the focused lens. Since the sapphire tube rather than the fiber is directly heated by the CO₂ laser, the passed fiber within the tube region can be homogeneously heated,which guarantee that the HLPGs can be robustly fabricated with a high yielding-rate [46]. In addition, the designed phase function $φ (z)$ shown in Fig. 1(a) and Fig. 2(a), respectively, were encoded into the local grating pitches, whose magnitudes can be expressed by

Λ_{j} = Λ_{0} (1 - Δ φ_{j} / 2 π), \begin{matrix}  \end{matrix} 1 \leq j \leq M,

where Λ₀ is the original pitch of the seed grating (i.e., the single-channel HLPG), which are adopted as 570 μm, and 540 μm for 3 channel and 9 channel HLPGs, respectively. M is the number of grating period adopted in one sampling period, and

Δ φ_{j} = φ_{j} - φ_{j - 1}

is the required phase change between the neighborhood grating’s period. By making use of the Eq. (10), we recalculated the local periods of the sampled HLPG. The corresponding pitches required for 3-channel and 9-channel HLPG in one sampling period (including 30 pitches of the HLPG) are shown in Fig. 5(a) and 5(b), respectively. From these two figures, it can be seen that due to the introduction of the phase-only sampling function, the grating pitches are no longer constant, which become variable at different local positions. During the fabrication processes, the local pitches was controlled by fine tuning the speeds of both the fiber-moving stage and the rotator. For example, one can continuously rotate the fiber at a speed of 10°/s by driving the rotator and meanwhile make the fiber moved through the sapphire tube at a constant speed of 15.833 μm/s, which was controlled by the translation stage 3 (as shown in Fig. 3). After the time of 36 seconds, one pitch grating with a length of 570 μm is then obtained. All the above procedures were pre-programmed and controlled through LabView software.

Fig. 4 Experimental setup for fabrication of multichannel HLPG.

Download Full Size | PDF

Fig. 5 Local pitches of the designed phase-only sampled HLPGs for (a) 3-channel and (b) 9-channel.

Download Full Size | PDF

Then we measured the transmission spectra of the fabricated multichannel HLPG with the same setup, where a white light source with wavelength ranging from 1350 to 1650 nm and an optical spectral analyzer (Anritsu MS9740A) were utilized. The measurement results are shown in Fig. 6, where the Fig. 6(a) shows the transmission spectrum of one typical 3-channel HLPG, whereas the Fig. 6(b) shows the transmission spectrum of one typical 9-channel HLPG. Noted that in order to cover the whole 9 channel in a wavelength band ranging from 1350 to 1650 nm, the central pitch of 540 μm for the 9-channel HLPG (corresponding to the central wavelength of 1477 nm) is particularly adopted, which is a little smaller than the magnitude of 570 μm utilized in Fig. 3(b). From Fig. 6(a), it can be seen that there really exist 3 notches (channels) near the central wavelength of 1560 nm and the channel spacing is about 30 nm, which agrees well with the simulation results as shown in Fig. 3(a). However, to compare with the simulation results, one can find that there exist some discrepancies in the notch depth for the resulted three channels (i.e., 7.5, 9.0, and 8.0 dB, respectively), the main reason to cause this non-uniformity probably be that the local pitches produced in the fabricated HLPG may not be exactly the same as those designed ones as shown in Fig. 5(a). From Fig. 6 (b), it can be seen that there exist 9 notches (channels) in the transmission spectrum. However, unlike the simulation results shown in Fig. 3(b) where the resulted 9 channel have almost an identical channel spacing and notch depth, the measured 9 channel shown in Fig. 6(b) show some non-uniformities in both of the channel spacing and the notch depth. In addition to the reason for causing the channel non-uniformity the Fig. 6(a) described above, the another possible reason to cause the discrepancy between the simulation results and the experimental ones may be that there may exist an overlap of the 9 channel (resulted from the particular LP₁₄ cladding mode in this study) with those ones resulted from the neighbor cladding modes LP₁₃ and LP₁₅, which could be resolved once if the channel spacing adopted is further decreased. Except for the above inconsistence, the experimental results almost agree with the theoretical ones, which in return means that the phase-only sampling method proposed in this study works well for both the LPGs and the HLPGs.

Fig. 6 Measuring results for transmission spectrum of the fabricated multichannel HLPGs with (a) 3-channel and (b) 9-channel.

Download Full Size | PDF

4. Conclusion

In this study, a new kind of multichannel HLPG based on the utilization of the phase-only sampling approach is firstly proposed and experimentally demonstrated, to the best of our knowledge. Both a 3-channel and 9-channel phase-only sampled HLPGs have been successfully fabricated by using the CO₂ laser writing technique. The proposed method allows one to fabricate a multi-channel HLPG with even a conventional single-mode (SM) fiber. The presented multichannel HLPGs may find potential applications to multi-wavelength OAM mode converters as well as the WDM sensors for simultaneous measurement of the torsion and the temperature.

Appendix

For convenience, a conventional LPG is considered here and its index change $Δ n_{s} (z)$ can be expressed as

Δ n_{s} (z) = R e {(Δ n_{0} / 2 \cdot e x p (i φ_{g})) e x p {i 2 π z / Λ_{0}}},

where z is the position along the grating, and

Δ n_{0}

,

Λ_{0}

, and

φ_{g} (z)

are the maximum index-modulation, central pitch, and the local phase of the grating, respectively. Noted that DC part of the index-change is neglected here. Characteristics of LPG for a particular cladding mode coupling with the core mode can be obtained by solving the following coupled mode equations:

\frac{d E_{A}}{d z} = i κ E_{B} \cdot e x p {i (2 β z)},

\frac{d E_{B}}{d z} = i E_{A} κ * \cdot e x p {- i (2 β z)},

where

β = (β_{A} - β_{B} - {2 π / Λ}_{0}) / 2

represents the wave number detuning,

E_{A}

and

β_{A}

,

E_{B}

and

β_{B}

represents the field amplitudes and propagation constants of the forward core- and cladding-mode, respectively.

κ

is complex coupling coefficient between these two modes, which represents the local grating amplitude and phase. By defining the local cross-transmission coefficient

ρ (z, β)

as

ρ (z, β) = \frac{E_{B} (z, β)}{E_{A} (z, β)},

then Eqs. (12) and (13) can be changed to the Riccati equation as:

\frac{d ρ (z, β)}{d z} = i κ * e x p {- i 2 β z} - i κ ρ^{2} e x p {i 2 β z},

when the grating is weak enough, i.e.,

{| ρ |}^{2} < < 1

(which is generally satisfied for each individual period of the grating), we roughly have

\frac{d ρ (z, β)}{d z} = i κ * e x p {- i 2 β z} .

Under the boundary conditions

ρ (0, β) = 0,

{| ρ |}^{2} < < 1

,

ρ (L, β) \approx t_{c} (β),

one can obtain the cross spectral response of the grating

t_{c} (β)

as

t {}_{c}{(β)} = i \int_{- \infty}^{\infty} κ (z) * e x p {- i 2 β z} d z .

The above equation obviously shows that the cross-transmission spectrum (corresponding to the field propagating in the cladding region) of the LPG has a Fourier transform relation with

κ (z) *

(which is proportional to the refractive index-change

Δ n (z)

). So for any desired spectrum

t_{c} (β)

of a LPG, the corresponding grating’s structure can be approximately obtained just by doing the Fourier transform directly.

Funding

Telecommunications Advancement Foundation; Nippon Sheet Glass Foundation for Materials Science and Engineering; International Exchange Program of NICT in Japan.

References

1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996). [CrossRef]

2. A. M. Vengsarkar, N. S. Bergano, C. R. Davidson, J. R. Pedrazzani, J. B. Judkins, and P. J. Lemaire, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21(5), 336–338 (1996). [CrossRef] [PubMed]

3. F. Chiavaioli, F. Baldini, S. Tombelli, C. Trono, and A. Giannetti, “Biosensing with optical fiber gratings,” Nanophotonics 6(4), 663–679 (2017). [CrossRef]

4. B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, and R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photonics Technol. Lett. 9(10), 1370–1372 (1997). [CrossRef]

5. D. S. Starodubov, V. Grubsky, and J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photonics Technol. Lett. 10(11), 1590–1592 (1998). [CrossRef]

6. O. Deparis, R. Kiyan, O. Pottiez, M. Blondel, I. G. Korolev, S. A. Vasiliev, and E. M. Dianov, “Bandpass filters based on π-shifted long-period fiber gratings for actively mode-locked erbium fiber lasers,” Opt. Lett. 26(16), 1239–1241 (2001). [CrossRef] [PubMed]

7. M. Harurnoto, M. Shigehara, and H. Suganurna, “Gain-flattening filter using long-period fiber gratings,” J. Lightwave Technol. 20(6), 1027–1033 (2002). [CrossRef]

8. S. Choi, T. Eom, Y. Jung, B. Lee, J. W. Lee, and K. Oh, “Broad-band tunable all-fiber bandpass filter based on hollow optical fiber and long-period grating pair,” IEEE Photonics Technol. Lett. 17(1), 115–117 (2005). [CrossRef]

9. B. J. Eggleton, R. E. Slusher, J. B. Judkins, J. B. Stark, and A. M. Vengsarkar, “All-optical switching in long-period fiber gratings,” Opt. Lett. 22(12), 883–885 (1997). [CrossRef] [PubMed]

10. M. Kulishov, D. Krcmarík, and R. Slavík, “Design of terahertz-bandwidth arbitrary-order temporal differentiators based on long-period fiber gratings,” Opt. Lett. 32(20), 2978–2980 (2007). [CrossRef] [PubMed]

11. V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996). [CrossRef] [PubMed]

12. H. J. Patrick, A. D. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. 16(9), 1606–1612 (1998). [CrossRef]

13. V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4(11), 457–466 (1999). [CrossRef] [PubMed]

14. L. Zhang, Y. Liu, L. J. Everall, J. A. R. Williams, and I. Bennion, “Design and realization of long-period grating devices in conventional and high birefringence fibers and their novel applications as fiber-optic load sensor,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1373–1378 (1999). [CrossRef]

15. S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14(5), R49–R61 (2003). [CrossRef]

16. X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensors utilizing long-period fiber gratings UV inscribed in D-fiber with enhanced sensitivity through cladding etching,” IEEE Photonics Technol. Lett. 16(5), 1352–1354 (2004). [CrossRef]

17. S. W. James, I. Ishaq, G. J. Ashwell, and R. P. Tatam, “Cascaded long-period gratings with nanostructured coatings,” Opt. Lett. 30(17), 2197–2199 (2005). [CrossRef] [PubMed]

18. I. Del Villar, F. J. Arregui, I. R. Matias, A. Cusano, D. Paladino, and A. Cutolo, “Fringe generation with non-uniformly coated long-period fiber gratings,” Opt. Express 15(15), 9326–9340 (2007). [CrossRef] [PubMed]

19. R. P. Murphy, S. W. James, and R. P. Tatam, “Multiplexing of fiber-optic long-period grating-based interferometric sensors,” J. Lightwave Technol. 25(3), 825–829 (2007). [CrossRef]

20. P. Pilla, P. Foglia Manzillo, M. Giordano, M. L. Korwin-Pawlowski, W. J. Bock, and A. Cusano, “Spectral behavior of thin film coated cascaded tapered long period gratings in multiple configurations,” Opt. Express 16(13), 9765–9780 (2008). [CrossRef] [PubMed]

21. Y. Wang, “Review of long period fiber gratings written by CO₂ laser,” J. Appl. Phys. 108(8), 081101 (2010). [CrossRef]

22. G. Rego, O. Okhotnikov, E. Dianov, and V. Sulimov, “High temperature stability of long-period fiber gratings produced using an electric arc,” J. Lightwave Technol. 19(10), 1574–1579 (2001). [CrossRef]

23. G. Humbert and A. Malki, “High performance bandpass filters based on electric arc-induced phase-shifted long-period fibre gratings,” Electron. Lett. 39(21), 1506–1507 (2003). [CrossRef]

24. S. Savin, M. J. F. Digonnet, G. S. Kino, and H. J. Shaw, “Tunable mechanically induced long-period fiber gratings,” Opt. Lett. 25(10), 710–712 (2000). [CrossRef] [PubMed]

25. G. Rego, J. R. A. Fernandes, J. L. Santos, H. M. Salgado, and P. V. S. Marques, “New technique to mechanically induce lone-period fibre gratings,” Opt. Commun. 220(1–3), 111–118 (2003). [CrossRef]

26. D. D. Davis, T. K. Gaylord, E. N. Glytsis, S. G. Kosinski, S. C. Mettler, and A. M. Vengsarkar, “Long-period fibre grating fabrication with focused CO₂ laser pulses,” Electron. Lett. 34(3), 302–303 (1998). [CrossRef]

27. Y. J. Rao, T. Zhu, Z. L. Ran, Y. P. Wang, J. Jiang, and A. Z. Hu, “Novel long-period fiber gratings written by high-frequency CO₂ laser pulses and applications in optical fiber communications,” Opt. Commun. 229(1–6), 209–221 (2004). [CrossRef]

28. F. Chiavaioli, F. Baldini, and C. Trono, “Manufacturing and spectral features of different types of long period fiber gratings: phase-shifted, turn-around point, internally tilted, and pseudo-random,” Fibers (Basel) 5(3), 29 (2017). [CrossRef]

29. Y. Liu, J. A. R. Williams, L. Zhang, and I. Bennion, “Phase shifted and cascaded long-period fiber gratings,” Opt. Commun. 164(1–3), 27–31 (1999). [CrossRef]

30. H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phase-shifted long-period fiber gratings,” IEEE Photonics Technol. Lett. 10(11), 1596–1598 (1998). [CrossRef]

31. Y. G. Han, B. H. Lee, W. Han, U. Paek, and Y. Chung, “Fibre-optic sensing applications of a pair of long-period fibre gratings,” Meas. Sci. Technol. 12(7), 778–781 (2001). [CrossRef]

32. R. P. Murphy, S. W. James, and R. P. Tatam, “Multiplexing of fiber-optic long-period grating-based interferometric sensors,” J. Lightwave Technol. 25(3), 825–829 (2007). [CrossRef]

33. S. W. James, I. Ishaq, G. J. Ashwell, and R. P. Tatam, “Cascaded long-period gratings with nanostructured coatings,” Opt. Lett. 30(17), 2197–2199 (2005). [CrossRef] [PubMed]

34. C. D. Pool, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9(5), 598–604 (1991). [CrossRef]

35. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004). [CrossRef] [PubMed]

36. O. V. Ivanov, “Fabrication of long-period fiber gratings by twisting a standard single-mode fiber,” Opt. Lett. 30(24), 3290–3292 (2005). [CrossRef] [PubMed]

37. S. Oh, K. R. Lee, U. C. Paek, and Y. Chung, “Fabrication of helical long-period fiber gratings by use of a CO₂ laser,” Opt. Lett. 29(13), 1464–1466 (2004). [CrossRef] [PubMed]

38. W. Shin, B. A. Yu, Y. C. Noh, J. Lee, D. K. Ko, and K. Oh, “Bandwidth-tunable band-rejection filter based on helicoidal fiber grating pair of opposite helicities,” Opt. Lett. 32(10), 1214–1216 (2007). [CrossRef] [PubMed]

39. L. Xian, P. Wang, and H. Li, “Power-interrogated and simultaneous measurement of temperature and torsion using paired helical long-period fiber gratings with opposite helicities,” Opt. Express 22(17), 20260–20267 (2014). [CrossRef] [PubMed]

40. G. Inoue, P. Wang, and H. Li, “Flat-top band-rejection filter based on two successively-cascaded helical fiber gratings,” Opt. Express 24(5), 5442–5447 (2016). [CrossRef] [PubMed]

41. C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period twisted elliptical fibers,” Appl. Opt. 51(10), C193–C197 (2012). [CrossRef] [PubMed]

42. H. Xu and L. Yang, “Conversion of orbital angular momentum of light in chiral fiber gratings,” Opt. Lett. 38(11), 1978–1980 (2013). [CrossRef] [PubMed]

43. G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. St. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012). [CrossRef] [PubMed]

44. C. Fu, S. Liu, Z. Bai, J. He, C. Liao, Y. Wang, Z. Li, Y. Zhang, K. Yang, B. Yu, and Y. Wang, “Orbital angular momentum mode converter based on helical long period fiber grating inscribed by hydrogen–oxygen flame,” J. Lightwave Technol. 36(9), 1683–1688 (2018). [CrossRef]

45. L. Fang and J. Wang, “Flexible generation/conversion/exchange of fiber-guided orbital angular momentum modes using helical gratings,” Opt. Lett. 40(17), 4010–4013 (2015). [CrossRef] [PubMed]

46. P. Wang and H. Li, “Helical long-period grating formed in a thinned fiber and its application to a refractometric sensor,” Appl. Opt. 55(6), 1430–1434 (2016). [CrossRef] [PubMed]

47. J. Li, G. Chen, P. Ma, L. P. Sun, C. Wu, and B. O. Guan, “Sampled Bragg gratings formed in helically twisted fibers and their potential application for the simultaneous measurement of mechanical torsion and temperature,” Opt. Express 26(10), 12903–12911 (2018). [CrossRef] [PubMed]

48. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef] [PubMed]

49. J. Bland-Hawthorn, M. Englund, and G. Edvell, “New approach to atmospheric OH suppression using an aperiodic fibre Bragg grating,” Opt. Express 12(24), 5902–5909 (2004). [CrossRef] [PubMed]

50. J. Bland-Hawthorn, S. C. Ellis, S. G. Leon-Saval, R. Haynes, M. M. Roth, H. G. Löhmannsröben, A. J. Horton, J. G. Cuby, T. A. Birks, J. S. Lawrence, P. Gillingham, S. D. Ryder, and C. Trinh, “A complex multi-notch astronomical filter to suppress the bright infrared sky,” Nat. Commun. 2(1), 581 (2011). [CrossRef] [PubMed]

51. F. Ouellette, P. A. Krug, G. Dhosi, T. Stephens, and B. Eggleton, “Broadband and WDM dispersion compensation using chirped sampled fibre Bragg gratings,” Electon. Lett. 31(11), 899–901 (1995). [CrossRef]

52. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photonics Technol. Lett. 10(6), 842–844 (1998). [CrossRef]

53. C. Wang, J. Azaña, and L. R. Chen, “Spectral Talbot-like phenomena in one-dimensional photonic bandgap structures,” Opt. Lett. 29(14), 1590–1592 (2004). [CrossRef] [PubMed]

54. J. Magné, P. Giaccari, S. LaRochelle, J. Azaña, and L. R. Chen, “All-fiber comb filter with tunable free spectral range,” Opt. Lett. 30(16), 2062–2064 (2005). [CrossRef] [PubMed]

55. R. Slavik, S. Doucet, and S. LaRochelle, “High-performance all-fiber Fabry-Perot filter with superimposed chirped Bragg grating,” J. Lightwave Technol. 21(4), 1059–1065 (2003). [CrossRef]

56. A. Buryak, J. Bland-Hawthorn, and V. Steblina, “Comparison of inverse scattering algorithms for designing ultrabroadband fibre Bragg gratings,” Opt. Express 17(3), 1995–2004 (2009). [CrossRef] [PubMed]

57. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-only sampled fiber Bragg gratings for high channel counts chromatic dispersion compensation,” J. Lightwave Technol. 21(9), 2074–2083 (2003). [CrossRef]

58. H. Li, M. Li, Y. Sheng, and J. E. Rothenberg, “Advances in the design and fabrication of high-channel-count fiber Bragg gratings,” J. Lightwave Technol. 25(9), 2739–2750 (2007). [CrossRef]

59. H. Li, M. Li, and J. Hayashi, “Ultrahigh-channel-count phase-only sampled fiber Bragg grating covering the S, C, and L bands,” Opt. Lett. 34(7), 938–940 (2009). [CrossRef] [PubMed]

60. H. Li and X. Chen, “High channel-count ultra-narrow comb-filter based on a triply sampled fiber Bragg grating,” IEEE Photonics Technol. Lett. 26(11), 1112–1115 (2014). [CrossRef]

61. X. Dong, N. Pan, P. Shum, and C. C. Chan, “Sampled long-period fiber grating filters with narrow stop bands,” Microw. Opt. Technol. Lett. 51(10), 2401–2403 (2009). [CrossRef]

62. S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983). [CrossRef] [PubMed]

63. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]

64. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber gratings filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997). [CrossRef]

Optimal design and fabrication of multichannel helical long-period fiber gratings based on phase-only sampling method

Abstract

1. Introduction

2. Principle of the sampled HLPG and the designed results

3. Fabrication of the phase-only sampled HLPG and the measurement results

4. Conclusion

Appendix

Funding

References

Cited By

Figures (6)

Equations (17)

Optics Express