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Abstract
Through analyzing the structure of the phase-shifted helical long-period fiber grating (HLPG), which is fabricated based on the thermally twisting method, we have shown that there exists a phase-diffusion effect when the thermal region is larger than the grating period itself, i.e., the inserted phase preset at particular period will be diffused to several neighboring periods, which causes a large distortion in the transmission spectrum. We have analytically proved that this kind of phase-diffusion effect can be quantified by doing the convolution between the preset phase function and the phase-diffusion function in spatial domain. According to the analytical results, we have proposed and successfully demonstrated a pre-compensation method to solve the phase diffusion effect. As an example, a phase-shifted HLPG with π phase-shift precisely inserted at middle position of the grating has been presented.
© 2017 Optical Society of America
1. Introduction
In the past decades, various kinds of long-period fiber gratings (LPGs) have been developed and widely used in the fields of optical communication, all-optical signal processing, and optical sensor etc [1–22]. Among which, the phase-shifted long-period fiber grating, i.e., the grating in which there exist one or more discrete phase jumps along the fiber axis, have recently attracted a great research interest attributed to its unique spectral characteristics, such as the broad band (several ten nanometers), high sensitivity to the ambient environment parameters [13–21].To date, various methods including either the UV post-processing, the post-etching technique, the CO2 laser technique, or by directly inserting a certain space into the grating during the fabrication have been proposed [22–28]. However, among all the above methods, the precise position and real magnitude of the phase-shift formed in a LPG have rarely been measured in experiment, which is, however, very important and strongly desirable whenever the phase-shifted LPG is utilized as either an all-optical signal processing or a high-sensitive sensing component, and especially when the phase-shifted LPG is used as a flat-top band-rejection filter, where several discrete phase-shifts to be precisely inserted at different positions of the LPG are strictly demanded [19–21],which could be a critical issue for practical fabrication.
On the other hand, as one kind of LPGs, helical long-period fiber grating (HLPG) have recently attracted a great research interest [29–39]. With intrinsic helicity characteristics, HLPG have been found many applications such as polarization controller, orbit-angular-momentum (OAM) mode converter, temperature and torque sensors, band-rejection filter etc. Recently, we have proposed and developed a novel technique to fabricate HLPG [38], where a sapphire tube is specially utilized instead of the commonly-used focal lens. A similar method has also been proposed to fabricate HLPG [35, 36], where either a micro oven or a micro-heater was employed instead of the CO2 laser. Among the above methods, since the heating region in which the fiber is periodically twisted is much larger than the grating period itself, it is extremely difficult to precisely produce a phase-shift at particularly position of a HLPG. Most recently, we have proposed and demonstrated a novel method to write a phase-shifted HLPG [28] based on CO2 laser method. However, the resulted phase-shift is an accumulated one (homogeneously distributed in a region larger than the heating region), strictly speaking, which is not the real one what we demanded.
In this study, by using the same setup as described in [38], firstly we fabricate the phase-shifted HLPGs where a π phase is presumed to insert at middle of the grating. Then we apply the microscopic imaging method (recently developed by us) [28] to observe and analyze the real structure of the produced phase-shifted HLPG. It is interesting for us to find that there exists a phase-diffusion effect, i.e., the inserted phase preset at particular period will be diffused to several neighboring periods with different phase magnitudes, which causes a large distortion in the transmission spectrum. We have analytically proved that this kind of phase-diffusion effect can be quantified by doing the convolution between the preset phase function and the phase-diffusion function in spatial domain. According to the analytical results, we have proposed and demonstrated a pre-compensated method, which enables to precisely produce a phase-shift at any preset position of the HLPG.
2. Principle scheme for the formation of a phase-shifted HLPG
The same fabrication setup of HLPG previously developed by us in [38] was utilized. For simultaneously creating a phase-shift in HLPG during its fabrication, we add an additional space at middle of the grating, and principle of which is depicted in Fig. 1(a). For comparison, the principle scheme to produce a conventional HLPG is also depicted in Fig. 1(b) where the grating period keeps a constant . From the Fig. 1(a), it is seen that any one magnitude of the inserted phase-shift could be obtained as long as the preset spacing P and the object phase satisfies the following equation,
Without losing any generalities, the phase-shift is assumed to be inserted at middle of the grating in this study. Moreover, if a phase-shift is inserted in a HLPG as shown in Fig. 1(a), then the local period at middle of the grating can be equivalently expressed as
Fig. 1 Principle scheme for the formation of a phase-shifted HLPG (a) Index-change profile with a phase-shift inserted, and (b) without phase-shift inserted.
3. Phase diffusion effect happened in a phase-shifted HLPG and its pre-compensation
A. Phase-diffusion effect happened in a phase-shifted HLPG
Based on the fabrication scheme shown in Fig. 1(a), a π phase-shifted HLPG which has a period of 640 μm and length of 23.04 mm (with 36 periods) is fabricated. Measurement results for the transmission spectrum is shown in Fig. 2. From this figure, it is easily seen that unlike the spectrum obtained in a conventional phase-shift LPG [27], the spectrum becomes non symmetrical, i.e., the resulted two dips have different depths, and the deeper one always lie in the longer wavelength side, these newly resulted discrepancies thus makes us believe that the really resulted phase may not lie exactly at the central position of the HLPG or the magnitude may not be the precise π, which probably be the reason why the spectral distortion appears. To verify the above, we further fabricated four kinds of phase-shifted HLPGs but with different phase-shifts of π, 1.5π, 3π, and 5π, respectively. Then we applied the microscopic imaging method [28] to observe and analyze both the inserted phase and the local periods through the whole grating. The calibrated results for the distribution of the four phase-shifted HLPGs are shown in Fig. 3 (a). Note that in Fig. 3 (a), both the grating position (the horizontal axis) and the phase magnitudes (vertical axis) all are normalized by the normal periodΛ. For easy visualization, the amplified parts of the Fig. 3 (a) lying between 12th and 24th periods are also shown in Fig. 3 (b).
Fig. 3 Measurement results for the phase distribution of the phase-shifted HLPGs vs the grating positions, where (a) within the whole length of the grating, and (b) the amplified parts within the central 12-period region of the HLPGs.
According to the fabrication scheme shown in Fig. 1 and the denotation in Eq. (2), for all four kinds of the HLPGs above, we do change the normal period Λ to period Λ1in order to achieve the pre-demanded phase-shift at the central position of the grating (i.e., right at the 18th period of the grating in our case), however as shown in Fig. 3(b) that, there actually exists a phase-diffusion phenomenon, i.e., the phase-changes appear not only at the central period of the grating (the 18th grating) but also at those periods between 16th and 20th, and each of them have different magnitudes. By summing up all the phase-changes appeared at the periods from 16th to 20th, it is interesting for us to find that the summation magnitude is exactly the same as the preset phase, which in return means that due to the phase-diffusion effect, the preset phase cannot be precisely inserted right at the middle of grating, however as an accumulated one, it can be inserted into the grating in about 5 grating periods in our case, this result is exactly the same as what we had obtained in [28] where an alternative approach to produce an accumulated phase in a HLPG had been proposed and experimentally demonstrated. From Fig. 3, it can also be found that no matter how different the four preset-phases are, the resulted phase distributions have nearly the same kind of profile except for the different amplitudes. Moreover, the diffusion region keeps a constant of about 3.2 mm (5 periods in our case), which is very close to the width of the heating region in our case. The above results implicate that the reason to produce the phase-diffusion effect (as shown in Fig. 3) is strongly related to the length of the heating region and could be regarded as a convolution result of the preset-phase with the phase diffusion function in spatial domain. Therefore, for the cases with a heating region of 10 mm in [36] and 3 mm in our case, the phase-diffusion effect cannot be neglected whenever the phase-shifted HLPG is fabricated.
B. Numerical simulation for modeling the phase-diffusion effect
As stated above, distribution of the diffused phases (as shown in Fig. 3) can be regarded as a convolution of the preset phase function with a basic phase diffusion function, which can be formulary expressed as
where represents preset phase function,represents the basic phase diffusion function, represents the convolution operation, and z is the position along the grating. The basic phase diffusion functioncan be numerically achieved just by fitting the phase diffusion data under the conditions of the different preset phases. The ideally preset phase-function can be expressed as:where represent the preset phase,represent a delta function, and represents the preset position where the inserted phase is demanded. Substituting the Eq. (4) in to Eq. (3), then we obtainFigure 4 shows the simulation results for the case of the π phase-shift HLPG, where Fig. 4(a) shows the preset phase function, it can be seen that a π phase-shift is preset at the middle of grating. Figure 4(b) shows the basic phase diffusion function, which is obtained by fitting the measured data in Fig. 3. Figure 4(c) shows the resulted phase distribution, which is achieved by using the Eq. (5). By further using the Eq. (2), all the data shown in Fig. 4(c) can be turned into the real resulted periods, which are shown in Fig. 4(d). Note that for easy visualization, only the parts the lies between 12thand 24th periods are shown in Fig. 4. Furthermore, by using the transfer matrix method (TMM) [40] and the practical data shown in Fig. 4 (d), the effect of the phase diffusion on transmission spectrum of a π phase-shift HLPG has been simulated, and the results are shown in Fig. 5, where all the unknown parameters are adopted the same as those what we used in the experiment of Fig. 2. From Fig. 5, it is seen that unlike the symmetric spectrum obtained in a conventional phase-shifted LPG [6, 26-27], the obtained spectrum is strongly distorted due to the phase-diffusion effect, and compared with the results shown in Fig. 2, it can be seen that the simulation results agree well with the experimental ones, which in return means that our assumptions about the key role of the heated region and the phase-diffusion model proposed above are reasonably correct, and will be useful to fabricate any kinds of the phase-shifted HLPGs.Fig. 4 Simulation results for the distribution of the phases and periods in a π phase-shifted HLPG, where (a) the objective phase, (b) the fitting result for phase-diffusion function, (c) the resulted phases, (d) the result periods.
C. An efficient method to compensate the phase-diffusion effect
Based on the analyses results given above, we have proposed an efficient method to compensate the phase-diffusion effect. First by doing the Fourier transform on both side of the Eq. (3), then we can obtain the following equation,
where represent the Fourier transform. From the above equation, it is easily seen that if the preset phases are adopted aswhere represents the inverse Fourier transform. Then after enduring the phase diffusion effect, the real resulted phase can be expressed asResult of the above equation means that by adopting the pre-compensated phases, we can precisely insert a phase at position of .Therefore, based on the above principle, if one wants to insert a π phase exactly at the middle of fiber grating, by using Eqs. (2) and (7), then the pre-compensated period distribution can be obtained, which is shown in Fig. 6. To verify the design results shown in Fig. 6, we fabricated a couple of π phase-shifted HLPGs where all the local gratings are precisely controlled to be exactly the same as those shown in Fig. 6. Figure 7 shows the transmission spectra of the fabricated π phase-shift HLPGs with and without pre-compensation, where “C” indicates the spectrum of one fabricated π phase-shift HLPG with pre-compensation whereas the “N C” indicates the one without pre-compensation (the same as shown in Fig. 2). To compare the results shown in Fig. 7 with those results obtained from a conventional π phase-shift LPG and shown in [6, 26-27], it can be seen that almost an ideal transmission spectrum of π phase-shift has been obtained, which in return means that the proposed pre-compensated phase method works pretty well.Fig. 7 Measurement results for transmission spectra of the fabricated π phase-shifted HLPGs with and without utilization of the pre-compensated phases.
4. Conclusion
Through analyzing the structure of a phase-shifted helical long-period fiber grating (HLPG), we have shown that there exists a phase-diffusion effect when the thermally twisting region is larger than the grating’s period itself, i.e., the inserted phase preset at particular period will be diffused to several neighboring periods with different phase magnitudes, which causes a large distortion in the transmission spectrum of the HLPG. We have analytically proved that this kind of phase-diffusion effect can be quantified by doing the convolution between the preset phase function and the phase-diffusion function in spatial domain. According to the analytical results, we have proposed and successfully demonstrated a pre-compensation method to solve the phase diffusion effect. As an example, a phase-shifted HLPG with π phase-shift precisely inserted at middle position of the grating has been presented.
Funding
Casio Science Promotion Foundation in Japan.
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