Spectral Characteristics of Square-Wave-Modulated Type II Long-Period Fiber Gratings Inscribed by a Femtosecond Laser

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Communication Spectral Characteristics of Square-Wave-Modulated Type II Long-Period Fiber Gratings Inscribed by a Femtosecond Laser

Xiaofan Zhao 1,2,†, Hongye Li 1,3,†, Binyu Rao 1,2, Meng Wang 1,2,3, Baiyi Wu 1,2,3 and Zefeng Wang 1,2,3,*

1 College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha 410073, China; [email protected] (X.Z.); [email protected] (H.L.); [email protected] (B.R.); [email protected] (M.W.); [email protected] (B.W.) 2 Hunan Provincial Key Laboratory of High Energy Laser Technology, Changsha 410073, China 3 State Key Laboratory of Pulsed Power Laser Technology, Changsha 410073, China * Correspondence: [email protected] † These authors contributed equally to this work.

Abstract: We study here the spectral characteristics of square-wave-modulated type II long-period fiber gratings (LPFGs) inscribed by a femtosecond laser. Both theoretical and experimental results indicate that higher-order harmonics refractive index (RI) modulation commonly exists together with the fundamental harmonic RI modulation in such LPFGs, and the duty cycle of a square wave has a great influence on the number and amplitudes of higher-order harmonics. A linear increase in the duty cycle in a series of square wave pulses will induce another LPFG with a minor difference in periods, which is useful for expanding the bandwidth of LPFGs. We also propose a method to reduce insertion loss by fabricating type II LPFGs without higher-order harmonic resonances. This work intensifies our comprehension of type II fiber gratings with which novel optical fiber sensors can be fabricated.

Citation: Zhao, X.; Li, H.; Rao, B.; Keywords: long-period fiber gratings; higher-order harmonics; femtosecond laser Wang, M.; Wu, B.; Wang, Z. Spectral Characteristics of Square-Wave- Modulated Type II Long-Period Fiber Gratings Inscribed by a Femtosecond Laser. Sensors 2021, 21, 3278. https:// 1. Introduction doi.org/10.3390/s21093278 LPFGs can realize the coupling between the copropagating core modes and cladding modes [1]. In the single-mode transmission region, the fundamental mode diverts to Academic Editor: Kyriacos Kalli cladding modes after passing through an LPFG, and LPFGs can also play a role as mode converters in few-mode fibers. Since the first demonstration in the 1990s, LPFGs have Received: 24 March 2021 been applied in many fields up to now, such as fiber sensors, fiber lasers, and so on [2]. Accepted: 7 May 2021 Due to the high sensitivity of coupled cladding modes to the environment, LPFGs found Published: 10 May 2021 their applications in RI sensing [3,4]. Furthermore, LPFGs were applied in temperature or stress detection [5–8]. Except for applications in fiber sensors, researchers applied Publisher’s Note: MDPI stays neutral LPFGs as spectral filtering devices in fiber lasers [9–12]. For example, LPFGs have been with regard to jurisdictional claims in used to suppress the stimulated Raman scattering (SRS) effect in high-power fiber lasers, published maps and institutional affil- where core-guided SRS light can be stripped out of the core by inserting LPFGs, and the iations. proportion of signal light in the output terminal can be increased [9–11]. By inscribing LPFGs in an erbium-doped fluoride glass fiber, Heck et al. proposed a new concept to mitigate the parasitic laser effect in mid-infrared fiber amplifiers [12]. Up to now, many methods have been utilized to fabricate LPFGs, including mechanical Copyright: © 2021 by the authors. stress [13,14], electrical arc [15,16], CO2 laser [2,17], femtosecond laser [18,19], and so Licensee MDPI, Basel, Switzerland. on. Exerting periodical mechanical stress on a few-mode fiber could induce coupling This article is an open access article between two copropagating core-guided modes, but such LPFGs could persist when distributed under the terms and mechanical stress was released [20]. A periodic taped structure induced by an electrical arc conditions of the Creative Commons or structuring a fiber with a CO2 laser could also realize the fabrication of LPFGs [21–24], Attribution (CC BY) license (https:// but these LPFGs were relatively fragile because a few sections of the fiber became thinner creativecommons.org/licenses/by/ just like microfibers [25–27]. UV lasers could inscribe relatively stable LPFGs by the 4.0/).

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direct-writing method or with the help of amplitude masks, but sufficient photosensitivity of fibers is necessary, so fibers should usually be hydrogen loaded in advance [28]. A femtosecond laser is another effective tool to fabricate LPFGs [29–31] with which the fiber is intact after fabrication, and the photosensitivity of fibers is not necessary. Recently, Heck et al. reported their investigations on type I LPFGs inscribed by a femtosecond laser, whose cross-RI modulation was positive (magnitude: 1 × 10–4) and took up the majority of the core area [32,33]. Type I LPFGs might degenerate in high-temperature environments. Another kind of LPFG inscribed by a femtosecond laser, namely type II LPFGs, is completely different. The RI profile is negative (magnitude: 1 × 10–3) and usually highly localized. Even when the temperature is more than 1000 ◦C, type II gratings can still keep their characteristics [34]. Their fine temperature tolerance makes LPFGs good candidates in high temperature sensing and high-power fiber laser systems. Square-wave- modulated type II LPFGs are easy to fabricate, but the insertion loss is relatively higher than that of type I, and the spectrum is in chaos. Up to now, few studies have interpreted their characteristics. In this paper, we investigate the characteristics of square-wave-modulated highly localized type II LPFGs inscribed by a femtosecond laser in detail. Both theoretical and experimental studies show that higher-order harmonic resonances coexist with fundamental frequency resonance in square-wave-modulated type II LPFGs, and the duty cycle of the square wave affects the amplitude and the number of higher-order harmonic resonances. With a linear increase in the duty cycle in a series of square wave pulses, resonances induced by a new LPFG of periods with little differences compared with the setting period occur. Based on the research results, a method to suppress higher-order harmonic resonances in type II LPFGs is proposed.

2. Theory LPFGs could realize the coupling between two different modes (core-guided or cladding-guided modes) transmitted in the same direction. The resonance wavelength is decided by a phase-matching condition:

2π β − β = m . (1) 1 2 Λ

Here, m is the resonant order, β1 and β2 are the propagating constant of the two coupling modes, and Λ is the grating period of the LPFG. According to the relation of β = 2πneff/λ, the resonant wavelength can be given by

Λ λ = (n − n ) . (2) e f f ,1 e f f ,2 m

Here, neff,1 and neff,2 are the effective RI of the two different modes. For single-mode operation, the core-guided fundamental mode can only be coupled to cladding modes, and a bunch of resonant wavelengths appear in the transmission spectrum. Considering the symmetry, the fundamental mode mainly couples to axisymmetric cladding modes LP0n. Figure1 shows the phase-matching condition of the ﬁrst-order resonance ( m = 1) when the grating period is 560 µm. The horizontal line represents 2π/Λ, and the intersection points deﬁne the resonant wavelengths. Simulation results indicate that LP01 mode couples to cladding mode LP02 at the wavelength of 1435 nm, to LP03 at 1475 nm, and LP04 at 1560 nm. Sensors 2021, 21, 3278 3 of 10 Sensors 2021, 21, x FOR PEER REVIEW 3 of 11

FigureFigure 1. 1.Phase-matching Phase-matching condition condition of of the the ﬁrst-order first-order harmonic harmonic resonance. resonance.

ExceptExcept for for the the resonant resonant points points of of LPFGs, LPFGs, the the RI RI modulation modulation function function is is another another focus focus thatthat has has to to be be investigated investigated in in detail. detail. For For most most femtosecond-laser-inscribed femtosecond-laser-inscribed LPFGs, LPFGs, the the RI RI modulationmodulation functionfunction alongalong thethe fiberfiber axisaxis isis inin thetheform formof of a a square square wave wave within within which which harmonicsharmonics cannotcannot bebe ignored, ignored, especially especially for for type type II II LPFGs. LPFGs. Each Each harmonic harmonic represents represents an an LPFGLPFG with with a a different different period. period. Moreover,Moreover, the the duty duty cycle cycle of of the the square square wave wave shows shows a a great great impactimpact onon the the amplitude amplitude of of harmonics. harmonics. InIn practice, practice, squaresquare wavewave modulation modulation cannotcannot be be infinitelyinfinitely longlong (provided(provided aa finite finite number number of of periods). periods). FigureFigure2 a2a shows shows a a typical typical square square wave.wave. Assuming Assuming n n is is the the number number of of periods, periods,Λ Λis is the the period, period, and and W W is is the the high high period, period, the the dutyduty cycle cycle is is defined defined as as the the ratio ratio of of high high period period to to the the total total period period Sensors 2021, 21, x FOR PEER REVIEW WW 4 of 11 D D= = .. (3) Λ Λ (3)

The expression of finite length square wave can be written as ×Λ −−−×ΛD N xn(1) Square() x=× A rect 2 .  ×Λ (4) n=1 D  Here, A is the amplitude of square wave. To study its spatial spectrum characteristics, Fourier transform is carried out on this function:

N (1)nD−×Λ+×Λ  ()Square() x=× A e− jkx dx (1)n−×Λ n=1 (5) (1−×−ee−ΛjkD ) (1 −Λ jNk ) =×A . jk×−(1 e−Λjk )

where k is the symbol of spatial frequency. Except k = 0, pole points of this function occur at k = 2πm/Λ (m = 1, 2, 3, 4, …), and the extremum of this function can also be found at these points. Figure 2b demonstrates the spatial spectrum of the square wave with different duty cycles when the number of the period is 72. It is obvious that not all harmonics coexist under any duty cycle, but it is not just fundamental harmonic that matters. For example, when the duty cycle is 50%, only odd-order harmonics can be excited, and the FigureFigure 2. 2.(a )( Schematica) Schematic of a square of a square wave. ( bwave.) Spatial (b spectrum) Spatial characteristics spectrum characteristics of a square wave of with a square wave amplitude of even harmonics is zero. The amplitude of each harmonic varies with the different duty cycles (color bar indicates the amplitude of each harmonic). withduty differentcycle. At least duty we cycles can say (color that bar in sqindicatesuare-wave-modulated the amplitude LPFGs, of each higher-order harmonic). harmonic RI modulation can also impact the transmission spectrum. Figure 3 shows the normalized amplitude of different harmonics versus the number of periods when the duty cycle is 50%. The amplitude of each harmonic increases linearly with the number of periods. The higher the order of the harmonic, the lower the amplitude. We can predict that in LPFG fabrication, the fundamental harmonic occurs even when the length of the LPFG is very short, but higher-order harmonics occur when the LPFG stretches to a certain length.

Figure 3. Amplitude of each resonance versus the number of periods.

Considering the situation where the duty cycle is no longer a constant in any part of a square wave sequence, the characteristics of the spatial spectrum show a large difference. For instance, the duty cycle grows linearly from 10% to 88.5% with the length of the square wave shown in Figure 4a (the total number of periods is 72): ×Λ Dn N xn−−−×Λ(1) =× 2 Squarel () x A rect  . (6) = D ×Λ n 1 n  =+ − Dnn 0.1 0.011( 1). (7)

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The expression of ﬁnite length square wave can be written as ! N x − D×Λ − (n − 1) × Λ Square(x) = A × rect 2 . (4) ∑ × n=1 D Λ

Here, A is the amplitude of square wave. To study its spatial spectrum characteristics, Fourier transform is carried out on this function:

N F( ( )) = × R (n−1)×Λ+D×Λ −jkx Square x A ∑ (n−1)×Λ e dx n=1 (5) ( − −jkDΛ)×( − −jNkΛ) = A × 1 e 1 e . jk×(1−e−jkΛ)

where k is the symbol of spatial frequency. Except k = 0, pole points of this function occur at k = 2πm/Λ (m = 1, 2, 3, 4, ... ), and the extremum of this function can also be found at these points. Figure2b demonstrates the spatial spectrum of the square wave with different duty cycles when the number of the period is 72. It is obvious that not all harmonics coexist under any duty cycle, but it is not just fundamental harmonic that matters. For example, when the duty cycle is 50%, only odd-order harmonics can be excited, and the amplitudeFigure 2. (a of) Schematic even harmonics of a square iszero. wave. The (b) Spatial amplitude spectrum of each characteristics harmonic varies of a square with wave the duty cycle.with different At least duty we cancycles say (color that bar in square-wave-modulatedindicates the amplitude of each LPFGs, harmonic). higher-order harmonic RI modulation can also impact the transmission spectrum. FigureFigure3 3shows shows the the normalized normalized amplitude amplitude of of different different harmonics harmonics versus versus the the number number ofof periodsperiods whenwhen thethe dutyduty cyclecycle isis 50%.50%. TheThe amplitudeamplitude ofof eacheach harmonicharmonic increasesincreases linearlylinearly withwith the the number number of of periods. periods. The The higher higher the the order order of theof the harmonic, harmonic, the lowerthe lower the amplitude.the ampli- Wetude. can We predict can predict that in that LPFG in fabrication,LPFG fabricat theion, fundamental the fundamental harmonic harmonic occurs evenoccurs when even thewhen length the length of the LPFGof the isLPFG very is short, very butshort, higher-order but higher-order harmonics harmonics occur when occur the when LPFG the stretchesLPFG stretches to a certain to a certain length. length.

FigureFigure 3.3.Amplitude Amplitude ofof eacheach resonanceresonance versusversus thethe numbernumber of of periods. periods.

ConsideringConsidering the the situation situation where where the the duty duty cycle cycle is is no no longer longer a constanta constant in in any any part part of of a squarea square wave wave sequence, sequence, the the characteristics characteristics of theof the spatial spatial spectrum spectrum show show a large a large difference. differ- Forence. instance, For instance, the duty the cycle duty grows cycle linearlygrows linearly from 10% from to 88.5%10% to with 88.5% the with length the of length the square of the wavesquare shown wave inshown Figure in4 aFigure (the total 4a (the number total ofnumber periods of isperiods 72): is 72):

N Dn×Λ ×Λ ! x− 2Dn − (n − 1) × Λ Square (x) = A × Nrect xn−−−×Λ(1). (6) l ∑ × =×n=1 D2n Λ Squarel () x A rect  . (6) = D ×Λ n 1 n Dn = 0.1 + 0.011(n − 1). (7) =+ − Dnn 0.1 0.011( 1). (7)

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Here, Dn is the duty cycle of the nth square wave pulse (n = 1, 2, …, 72). The Fourier transform of Equation (4) can be written as:

N (1)nD−×Λ+×Λ ()=× n − jkx  Squarel () x A e dx (1)n−×Λ n=1 (8) (1−×−−×−×−eeeee−Λjk1.011 ) (1 −Λ−Λ jNk ) jk 0.1 (1 −Λ jk ) (1 − jNk 1.011 Λ ) =×A jk×−(1 e−Λjk ) ×− (1 e − j1.011 k Λ )

In Equation (6), there are two series of pole points except k = 0. The first pole points series is represented as k = 2πm/Λ, and the second is k = 2πm/1.011Λ (m = 1, 2, 3, …). The spatial spectrum characteristics are demonstrated in Figure 4b. When the number of periods is 72 (full length), two series of harmonics occur in the spatial spectrum, and as the order of harmonics increases, the frequency difference also grows. Only two fundamental harmonic frequencies overlap with each other for the reason of small frequency differences. However, if the number of the period is half (36 periods), although two series of resonance frequencies are still obvious in higher-order harmonics, the spatial frequency difference is extremely small in the first- and second-order resonance region. In practical LPFG inscription, if the duty cycle increases with the length of the LPFG, one possible Sensors 2021, 21, 3278 5 of 10 phenomenon we can observe is that another comb of dips grows up near the original second harmonic resonance wavelengths, and the bandwidth expands consequently.

Figure 4. (a) Waveform of a square wave wave with with a a linearly linearly growing growing duty duty cycle. cycle. (b (b) )Spatial Spatial spectrum spectrum characteristics of a square wavewave withwith aa linearlylinearly growinggrowing dutyduty cycle.cycle.

3. ExperimentHere, Dn is the duty cycle of the nth square wave pulse (n = 1, 2, ... , 72). The Fourier transformA schematic of Equation inscription (4) can diagram be written for as:square-wave-modulated LPFGs is shown in Fig- ure 5a. A femtosecond laser (Pharos Light Conversion) with 190 fs pulses, a central wave- N length of 515 nm, andF( a repetition( )) = rate× of 1 kHzR (n− 1s) ×focusedΛ+Dn×Λ with−jkx an oil-free microscope Squarel x A ∑ (n−1)×Λ e dx objective (Mitutoyo) with a magnificationn of=1 100×. The pulse energy after the objective(8) is ( − −j1.011kΛ)×( − −jNkΛ)− −j0.1kΛ×( − −jkΛ)×( − −j1.011NkΛ) = A × 1 e 1 e e 1 e 1 e . about 150 nJ. An SMF (the core diameterjk×(1− ise− 8.2jkΛ) ×μ(m,1− eand−j1.011 thekΛ) numerical aperture is 0.14) is fixed on a high-precision three-axis linear translation stage. During LPFGs fabrication, the horizontalIn Equation speed (6),v is thereset as are1000 two μm/s, series and of the pole spacing points between except k two= 0. successive The ﬁrst pole points points is 1 seriesμm in istheory. represented However, as k =the 2π cylindricalm/Λ, and thelens second effect isofk fiber= 2π mcannot/1.011 beΛ ignored(m = 1, 2, in 3, an... oil-free). The spatialinscription spectrum environment, characteristics the RI are modulation demonstrated generated in Figure by4b. aWhen pulsethe takes number on an of ellipsoid periods iswith 72 (fullits major length), axis two aligning series in of the harmonics horizontal occur direction in the spatial of fiber, spectrum, as shown and in Figure as the order 5b. The of harmonicslength of the increases, major axis the frequencyis more than difference 3 μm when also grows. the pulse Only energy two fundamental is 150 nJ; thus, harmonic the RI frequencies overlap with each other for the reason of small frequency differences. However, if the number of the period is half (36 periods), although two series of resonance frequencies are still obvious in higher-order harmonics, the spatial frequency difference is extremely

small in the ﬁrst- and second-order resonance region. In practical LPFG inscription, if the duty cycle increases with the length of the LPFG, one possible phenomenon we can observe is that another comb of dips grows up near the original second harmonic resonance wavelengths, and the bandwidth expands consequently.

3. Experiment A schematic inscription diagram for square-wave-modulated LPFGs is shown in Figure5a. A femtosecond laser (Pharos Light Conversion) with 190 fs pulses, a central wavelength of 515 nm, and a repetition rate of 1 kHz s focused with an oil-free microscope objective (Mitutoyo) with a magnification of 100×. The pulse energy after the objective is about 150 nJ. An SMF (the core diameter is 8.2 µm, and the numerical aperture is 0.14) is fixed on a high-precision three-axis linear translation stage. During LPFGs fabrication, the horizontal speed v is set as 1000 µm/s, and the spacing between two successive points is 1 µm in theory. However, the cylindrical lens effect of fiber cannot be ignored in an oil-free inscription environment, the RI modulation generated by a pulse takes on an ellipsoid with its major axis aligning in the horizontal direction of fiber, as shown in Figure5b. The length of the major axis is more than 3 µm when the pulse energy is 150 nJ; thus, the RI modulation generated by two successive pulses overlaps with each other, as shown in Figure5c. Because the LPFG is highly localized, and if the refractive index modulation Sensors 2021, 21, x FOR PEER REVIEW 6 of 11

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modulation generated by two successive pulses overlaps with each other, as shown in Figure 5c. Because the LPFG is highly localized, and if the refractive index modulation generatedgenerated by by a femtoseconda femtosecond laser laser is notis not high high enough, enough, the spectrumthe spectrum will notwill appear. not appear. We tried We ◦ totried place to ourplace LPFGs our LPFGs in a high-temperature in a high-temperature (500 C) (500 furnace °C) furnace for more for than more 8 h. than The spectra8 h. The afterspectra cooling after showedcooling showed little difference little difference compared compared to the spectrato the spectra before before high-temperature high-temper- annealing.ature annealing. Thus, weThus, conﬁrm we confirm the LPFGs the LPFGs are type are II. type II.

FigureFigure 5. 5.( a(a)) Schematic Schematic of of LPFG LPFG fabrication. fabrication. (b (b) RI) RI modulation modulation generated generated point-by-point point-by-point (magniﬁca- (magnifi- tion:cation: 100 100×).×). (c )(c Microscope) Microscope image image of of a square-wave-modulateda square-wave-modulated LPFG LPFG (magniﬁcation: (magnification: 100 100×).×).

TheThe growth growth process process of of an an LPFG LPFG is is illustrated illustrated in in Figure Figure6 .6. The The period period is is 560 560µ μm,m, and and thethe total total lengthlength isis 40,32040,320 μµm. The The spectra spectra of of five ﬁve different different lengths lengths are are recorded recorded in inour our ex- experiment.periment. As As we we discussed discussed in Figure in Figure 3, only3, only the LPFG the LPFG stretches stretches a certain a certain length length can higher- can higher-orderorder harmonic harmonic resonance resonance of the ofgrating the grating occur. occur. In our In experiment, our experiment, when when the length the length of the ofLPFG the LPFG is longer is longer than than20,160 20,160 μm, µthem, higher-order the higher-order harmonic harmonic resonance resonance of the of the grating grating be- becomescomes discernible. discernible. With With the the increase increase in in the the LPFG length, thethe insertioninsertion lossloss goesgoes up, up, as as scatteringscattering lossloss induced induced by by a a femtosecond femtosecond laser laser grows grows up up in in this this process. process. The The coupling coupling betweenbetween the the fundamentalfundamental modemode and higher-order cladding mode mode (LP (LP0404) )saturates saturates prior prior to tothe the lower-order lower-order cladding cladding mode mode (such (such as LP as03 LP and03 andLP02 LP). Because02). Because the LPFG the LPFGdoes not does strictly not strictlylocalize localize in the center in the of center the core, of the the core,cross-RI the profile cross-RI takes proﬁle on an takes axisymmetric on an axisymmetric rather than rathera circularly than a symmetric circularly form; symmetric thus, birefringence form; thus, birefringence becomes obvious becomes when obvious the LPFG when is long. the LPFGIn the is LPFG long. fabrication In the LPFG process, fabrication mode process,coupling mode is a dynamic coupling process, is a dynamic for which process, energy for in whichcladding energy modes in claddingmay convert modes back may to the convert fundamental back to the mode fundamental when the grating mode when reaches the a gratingcertain reaches length (this a certain process length is named (this process overcoupli is namedng). overcoupling).As shown in Figure As shown 6, after in Figure the over-6 , aftercoupling the overcoupling process, the resonance process, the re-enhances, resonance and re-enhances, birefringence and occurs birefringence in the meantime. occurs in the meantime.

Figure 6. Spectrum evolution during LPFG fabrication (period: 560 μm; duty cycle: 50%. Insertion loss from 1200 to 1350 nm increases with the grating length). Figure 6. Spectrum evolution during LPFG fabrication (period: 560 µm; duty cycle: 50%. Insertion loss from 1200 to 1350 nm increases with the grating length).

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Figure7 illustrates the transmission spectra of LPFGs with different duty cycles. The lengthFigure of LPFGs 7 illustrates is ﬁxed the as transmission 40,320 µm, and spectr thea of period LPFGs of with LPFGs different is 560 dutyµm. cycles. Thus, theThe numberlength of of LPFGs periods is isfixed 72. Exceptas 40,320 for μ them, ﬁrst-orderand the period resonance, of LPFGs where is 560 the μ fundamentalm. Thus, the mode num- couplesber of periods to the LPis 0272., LPExcept03, and for LP the04 claddingfirst-order mode resonance, at around where 1425, the 1480,fundamental and 1550 mode nm, respectively,couples to the which LP02, isLP close03, and to LP the04 theoreticalcladding mode analysis at around in Figure 1425,1, and1480, the and error 1550 mainly nm, re- comesspectively, from which the neglect is close of materialto the theoretical dispersion, analysis the higher-order in Figure 1, harmonic and the resonanceerror mainly of thecomes grating from isthe also neglect visible. of material The occurrence dispersion of, the higher-order higher-order harmonic harmonic resonance resonance creates of the chaosgrating over is also the visible. transmission The occurrence spectrum of of higher-order the LPFG. What harmonic has to resonance be pointed creates out is chaos that theover higher-order the transmission resonance spectrum generated of the LPFG. by the What LPFG has with to be a dutypointed cycle out of is 50%that the is weaker higher- thanorder the resonance other two generated LPFGs, whichby the canLPFG be explainedwith a duty by cycle what of we 50% discussed is weaker in than Figure the2 otherthat thetwo harmonic LPFGs, which number can ofbe squareexplained waves by what with we a duty discussed cycle ofin 50%Figure is less2 that than the theharmonic other dutynumber cycle. of square waves with a duty cycle of 50% is less than the other duty cycle.

FigureFigure 7.7. TransmissionTransmission spectra spectra of of LPFGs LPFGs with with different different duty duty cycles cycles (insertion (insertion loss loss of of the the LPFG LPFG with a dutywith cyclea duty of cycle 10% isof the10% smallest, is the smallest, and that and of thethat LPFG of the with LPFG a dutywith cyclea duty of cycle 25% isofthe 25% highest). is the highest). Figure8 demonstrates the transmission spectra of different-order resonances. The periodsFigure of the 8 LPFGsdemonstrates are 560, the 1120, transmission and 1680 µ spm,ectra which, of different-order respectively, correspond resonances. to theThe first-,periods second-, of the andLPFGs third-order are 560, harmonic1120, and resonances1680 μm, which, generated respectively, by grating correspond in the focusing to the wavelengthfirst-, second-, band. and The third-order grating length harmonic is 40,320 resoµnancesm regardless generated of the by period.grating Whenin the thefocusing duty cyclewavelength is 50%, band. the second-order The grating length harmonic is 40,320 resonance μm regardless of the grating of the disappears period. When as thethe blueduty linecycle in is Figure 50%, 8the shows, second-order but the second-order harmonic reso harmonicnance of resonance the grating of disappears the grating isas evidentthe blue ifline the in duty Figure cycle 8 shows, is 25% but (red the line second-order in Figure8). harmonic The pink resonance line performs of the thegrating transmission is evident spectrumif the duty when cycle the is period25% (red is 1680line inµm Figure and the 8). dutyThe cyclepink line is 50%. performs The resonant the transmission intensity isspectrum relatively when weak the compared period is 1680 to the μm other and situation,the duty cycle as the is amplitude50%. The resonant of the third-order intensity is harmonicrelatively resonanceweak compared is only to one-third the other of situat whation, it is as in the amplitude first-order resonance.of the third-order har- monicTo resonance testify the is inferenceonly one-third in Figure of what4, an it LPFGis in the with first-order different resonance. duty cycles (linearly increasing from 10% to 90%) in different periods is fabricated. Because the frequency difference is relatively small in the first-order resonance region, the period of the grating is set as 1120 µm to realize the second-order harmonic resonance of the grating in the focusing wavelength band, and the total length of the LPFG is 80,640 µm. Figure9 illustrates the transmission spectrum of the LPFG during our fabricating process. When the grating length is 40,320 µm (half of the total length), the spectrum is similar to that depicted in Figure6, where the second-order harmonic resonance of the grating takes up the majority, and the higher-order harmonic resonance of the grating shows up as well. However, the spectrum of the full-length LPFG is much more different. Except for the former resonant points, another series of dips (marked as blue points) appear. Meanwhile, the birefringence (marked as green points) also presents for the reason of the long grating length. These phenomena agree well with the aforementioned theoretical analysis. A derived LPFG

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that possesses different periods compared with the setting one is fabricated by linearly Sensors 2021, 21, x FOR PEER REVIEW 8 of 11 increasing the duty cycle of the time series pulse. The method shows the potential in bandwidth widening.

FigureFigure 8.8. Transmission spectra of LPFGs LPFGs with with periods periods of of 560, 560, 1120, 1120, and and 1680 1680 μmµm (grating (grating length: length: 40,320 µμm. Resonant intensity decreases with the period, and when the duty cycle is 50%, the sec- Sensors 2021, 21, x FOR PEER REVIEW40,320 m. Resonant intensity decreases with the period, and when the duty cycle is 50%,9 of the 11 second-orderond-order harmonic harmonic resonance resonance of ofthe the grating grating disappears). disappears).

To testify the inference in Figure 4, an LPFG with different duty cycles (linearly increasing from 10% to 90%) in different periods is fabricated. Because the frequency difference is relatively small in the first-order resonance region, the period of the grating is set as 1120 μm to realize the second-order harmonic resonance of the grating in the focusing wavelength band, and the total length of the LPFG is 80,640 μm. Figure 9 illustrates the transmission spectrum of the LPFG during our fabricating process. When the grating length is 40,320 μm (half of the total length), the spectrum is similar to that depicted in Figure 6, where the second-order harmonic resonance of the grating takes up the majority, and the higher-order harmonic resonance of the grating shows up as well. However, the spectrum of the full-length LPFG is much more different. Except for the former resonant points, another series of dips (marked as blue points) appear. Meanwhile, the birefringence (marked as green points) also presents for the reason of the long grating length. These phenomena agree well with the aforementioned theoretical analysis. A derived LPFG that possesses different periods compared with the setting one is fabricated by linearly increasing the duty cycle of the time series pulse. The method shows the potential in bandwidth widening. The coupling characteristics of square-wave-modulated type II LPFGs are much more complex compared to those of type I, which provides more choices for the fiber FigureFiguresensing 9. system.SpectrumSpectrum However, evolution evolution theduring during existence LPFG LPFG fabrication of fabricationhigher-order (insertion (insertion harmonic loss lossincreases resonance increases with withtheof the grating the grating grat- inglength).disturbs length). the transmission spectrum and creates undesirable insertion loss, which forbids type II LPFGs as the fiber laser system application. Thus, a method to suppress the higher- 4.order ConclusionsThe harmonic coupling resonance characteristics of the ofgrating square-wave-modulated in type II LPFGs is typenecessary. II LPFGs Considering are much morethat a complexsquareWe wave comparedinvestigated is composed to thosethe ofcoupling of several type I, characteristicsharmon which providesics and ofhigher-order more square-wave-modulated choices harmonics for the ﬁber RI modula- sensingtype II system.LPFGstion contributes fabricated However, to theby higher-order a existence femtosecond of harmonic higher-order laser. Both resoharmonic nancestheoretical of resonancethe and grating, experimental of if the the grating RI results modulation disturbs show thethatalong transmission except with thefor fiberthe spectrum fundamental axis takes and on createsfrequency a sinusoidal undesirable re sonanceshape, insertionthe induced higher-order loss, by a whichsquare harmonic forbids wave, resonance higher- type II LPFGsorderof the harmonic asgrating the ﬁber cannot resonances laser occur. system are To application. alsorealize signific our Thus,antassumption, in a such method LPFGs. the to repetition suppress The duty thefrequency cycle higher-order shows of the a harmonicgreatfemtosecond effect resonance on laser the amplitude should of the be grating andhigh number enough in type of to higher-order II make LPFGs the is RI necessary.harmonic modulation resonances Considering area generated of LPFGs. that by a squareBytwo varying successive wave the is composed dutypulses cycle overlap of in several a serieswith harmonics eachof square other. and wave The higher-order pulsepulses, energy an harmonicsLPFG is a with time-varying RI two modulation different sine contributesperiodswave, and compared tothe higher-order period to theof thefabricated harmonic sine wave period resonances equals can the be of period thederived, grating, of bythe ifwhich LPFG. the RI we In modulation thiscan expandway, alongthe the RI bandwidthmodulation of along the LPFG. the fiber The axis research takes onwork a si enne-likehances shape our comprehension(RI modulation isof basedtype II on fiber the gratingnonlinear and multiphoton shows potential absorption in novelty process, sensor and fabrication.the RI modulation A method along to the fabricate fiber axis type can- II LPFGsnot be witha strict a higher-ordersine-wave). Theharmonic higher-order resonance harmonic of the gratingresonance suppression of the grating is proposed. may be Inlargely the future, suppressed we will by realize this method. the fabrication of sine-wave-modulated type II LPFGs. In this way, the higher-order harmonic resonance of the grating can be avoided, and the insertion loss can be reduced. Type II grating is more suitable for applications of optical fiber sensors and fiber laser systems.

Author Contributions: X.Z. investigation, data curation, writing—review and editing; H.L. investigation, data curation, writing—original draft preparation; B.R. software; M.W. methodology, funding acquisition; B.W. validation; Z.W. supervision, project administration. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Outstanding Youth Science Fund Project of Hunan Prov- ince Natural Science Foundation (2019JJ20023) and National Natural Science Foundation of China (NSFC) (11974427, 12004431), and State Key Laboratory of Pulsed Power Laser (SKL-2020-ZR05). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data sharing not applicable. Conflicts of Interest: The authors declare no conflicts of interest.

References 1. Vengsarkar, A.M.; Lemaire, P.J.; Judkins, J.B.; Bhatia, V.; Erdogan, T.; Sipe, J.E. Long-period fiber gratings as band-rejection filters. J. Light. Technol. 1996, 14, 58–65, doi:10.1109/50.476137. 2. Wang, Y. Review of long period fiber gratings written by CO2 laser. J. Appl. Phys. 2010, 108, 081101, doi:10.1063/1.3493111. 3. Patrick, H.J.; Kersey, A.D.; Bucholtz, F. Analysis of the response of long period fiber gratings to external index of refraction. J.

Sensors 2021, 21, 3278 9 of 10

with the fiber axis takes on a sinusoidal shape, the higher-order harmonic resonance of the grating cannot occur. To realize our assumption, the repetition frequency of the femtosecond laser should be high enough to make the RI modulation area generated by two successive pulses overlap with each other. The pulse energy is a time-varying sine wave, and the period of the sine wave equals the period of the LPFG. In this way, the RI modulation along the fiber axis takes on a sine-like shape (RI modulation is based on the nonlinear multiphoton absorption process, and the RI modulation along the fiber axis cannot be a strict sine-wave). The higher-order harmonic resonance of the grating may be largely suppressed by this method.

4. Conclusions We investigated the coupling characteristics of square-wave-modulated type II LPFGs fabricated by a femtosecond laser. Both theoretical and experimental results show that except for the fundamental frequency resonance induced by a square wave, higher-order harmonic resonances are also significant in such LPFGs. The duty cycle shows a great effect on the amplitude and number of higher-order harmonic resonances of LPFGs. By varying the duty cycle in a series of square wave pulses, an LPFG with two different periods compared to the fabricated period can be derived, by which we can expand the bandwidth of the LPFG. The research work enhances our comprehension of type II fiber grating and shows potential in novelty sensor fabrication. A method to fabricate type II LPFGs with a higher-order harmonic resonance of the grating suppression is proposed. In the future, we will realize the fabrication of sine-wave-modulated type II LPFGs. In this way, the higher-order harmonic resonance of the grating can be avoided, and the insertion loss can be reduced. Type II grating is more suitable for applications of optical fiber sensors and fiber laser systems.

Author Contributions: X.Z. investigation, data curation, writing—review and editing; H.L. investigation, data curation, writing—original draft preparation; B.R. software; M.W. methodology, funding acquisition; B.W. validation; Z.W. supervision, project administration. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Outstanding Youth Science Fund Project of Hunan Province Natural Science Foundation (2019JJ20023) and National Natural Science Foundation of China (NSFC) (11974427, 12004431), and State Key Laboratory of Pulsed Power Laser (SKL-2020-ZR05). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data sharing not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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