We demonstrate theoretically and experimentally that it is feasible to draw the microstructured fiber with longitudinally varying diameter (FLVD) whose diameter varies sharply in a short fiber length. It is elucidated that during the fiber drawing process the tension is linearly proportional to the natural logarithm of the fiber drawing speed. As a result, the tension is not so sensitive to the fiber diameter. Moreover, this sensitivity can be decreased by using a large diameter ratio of preform to fiber. Owing to the low sensitivity the FLVD with diameter varying sharply in a short fiber length can be drawn directly from the preform. Additionally we show that the microstructural geometry of FLVD does not depend on the varying diameter. The deformation in microstructural geometry is determined by the fiber segment with the smallest diameter. We fabricate a FLVD of which the diameter decreases by 75% in a fiber length of 10 cm. By using this fiber we demonstrate the 600-1800 nm supercontinuum (SC) generation and the 532 nm second harmonic generation pumped by a picosecond fiber laser. The SC spectra by the conventional fibers with the largest and the smallest diameters of the FLVD are also shown, respectively. The comparisons show that the FLVD has the broadest SC spectrum due to its high nonlinearity, varying dispersion, and high damage threshold.
© 2012 OSA
Optical fiber with longitudinally varying diameter (FLVD) has some unique characteristics which involve the longitudinally varying chromatic dispersion and nonlinearity. Highly nonlinear FLVDs have already found applications in supercontinuum (SC) generation, pulse compression, soliton propagation, sensor, coupler, etc [1–7]. Recently novel phenomena such as soliton blueshift have been found in highly nonlinear microstructured FLVD . Generally this kind of fiber can be fabricated by tapering. The process is as follows: A segment of fiber is selected. Both tips of the fiber are fixed in an elongation device with heating source. The heating source can be either a laser beam, a tube shaped furnace, or a flame brush [9,10]. The fiber is heated to the softening temperature and then is elongated gradually. There are some disadvantages for this technique. The length of the tapered segment is restrained by the elongation device and the fiber used. It is usually in the magnitude from several centimeters to several tens of centimeters. The tapering process is time-consuming and unstable. Especially when tapering the fiber to have a diameter several times smaller than the original, the tension often varies dramatically, and the fiber is easy to break (the reason will be discussed in the section of 2.1). Such a technique might have a poor cost-effectiveness for the commercial manufacture.
Another option to fabricate FLVD is to draw it directly from the preform at the varying fiber drawing speed. At a given feeding speed of preform, the diameter of fiber can be varied by modulating the fiber drawing speed. The technique omits tedious step of fiber reprocessing. The fiber length can be as long as conventional fiber. By modulating the fiber drawing speed periodically, many FLVDs can be obtained by cutting the fiber fabricated in one time of fiber drawing. FLVDs by this technique have already been demonstrated before. Reference  shows a fiber with diameter gradually decreases from 125 μm to 80 μm in a fiber length of 100 m. Reference  presents a fiber with a diameter varying periodically. The varying period is 160 m. The maximum varying ratio is 0.06%.
However, there are two limitations for the present FLVDs fabricated by the method of directly drawing. One is the diameter varies too slowly along the fiber length. For the fiber with high nonlinearity pumped by ultrashort pulse, both the nonlinear length and dispersion length are often no more than several meters, and many are in the magnitude of several centimeters or even shorter. Usually fiber with length in this magnitude is already enough for practical applications. However, the diameter of FLVD by directly drawing varies so slowly that it is almost constant in the characteristic lengths. The other limitation is that the total variation in diameter is too limited. The diameter ratio of the maximum to the minimum (DRMM) is usually less than 2. Consequently the variations in dispersion and nonlinearity are too limited to meet the practical applications, especially in some cases where the decreasing diameter is expected to greatly improve the nonlinearity . According to the above explanation, currently an important problem need to be addressed for this technique is the feasibility of a large variation in the fiber diameter in a short fiber length. Additionally, during the fiber drawing process of FLVD, a varying fiber drawing speed is necessary. We need to know the influence of the large variation in fiber drawing speed on the shape of microstructure. Then we can know how the geometry of microstructure depends on the fiber diameter.
In this work, we try to address these two issues. Theoretical analyses and experiments show that, through the method of directly drawing FLVD from preform, it is feasible to realize a much larger DRMM in a much shorter fiber length compared with those which have been demonstrated before, and the geometry of the microstructure can be independent on the varying diameter. By using a preform with large holes we demonstrate the FLVD with a DMMR of 4 in a fiber length of only 10 cm. By using the fabricated FLVD we obtained broad SC generation and second harmonic generation (SHG) pumped by picosecond pulse.
2. Theoretical analyses
2.1 The feasibility to directly draw FLVD with diameter varying sharply in a short fiber length
During the fiber-drawing process, the diameter of fiber is determined by:
In Eq. (1) d is the diameter of fiber. D is the diameter of preform. Vf and Vp are the fiber drawing speed and the preform feeding speed, respectively. Conventionally, Vp and D are constant, since their variations can result in severe variations in the fiber drawing conditions. Therefore, d is usually varied by changing Vf. The degree of variation in Vf determines the degree of variation in d. We need to know how much extent Vf can be changed.
The maximum extent of variation in Vf depends on its influence on the fiber drawing tension. If the tension changes dramatically with the variation of Vf, it is difficult to fabricate FLVD with diameter varying greatly in a short fiber length, because the tension is easy to exceed the limitation that the fiber can endure, and the fiber is subject to break before being coated. In the following discussion we will analyze how the variation in tension depends on the variation in Vf.
Figure 1 shows a schematic diagram of the neck region of a preform. C1 corresponds to the cross section where the diameter is the same as that of the fiber. C2 corresponds to the cross section where the diameter is the same as that of the preform. The neck region between C1 and C2 is the transition section. For a fiber drawing process, according to the Newtonian flow model the following relationship exists :
In Eq. (2) F is the tension, η is the viscosity, z is the position of d at the axis of preform. Here the origin of z locates at C1. At C1, z = 0. At C2, z = l. Combining Eq. (1) with Eq. (2) we can get:
Taking parameters at C1 and C1 as the boundary conditions, we can obtain:
In Eq. (4) except for F and Vf other parameters can be taken as constant, so F is linearly proportional to the natural logarithm of Vf. It is interesting because F is not very sensitive to Vf. Therefore it is possible to adjust Vf to a large extent without breaking the fiber. Suppose that Vf is increased to NVf (N is a multiple), and F is increased to F’, we can get Eq. (5):
In Eq. (5) ρ is the proportion of variation in F. It is inversely proportional to ln(Vf/Vp). A large ratio of Vf to Vp is advantageous to decrease the sensitivity of tension to fiber drawing speed. Combining Eq. (5) with Eq. (1), we can get:
From Eq. (6) we can know a preform with comparatively large diameter is helpful to decrease ρ. D/d usually can be around 100 or much larger. Given D/d = 100, for the N with a value of several tens, ρ is less than 50%, so we can reduce the fiber diameter to d/ without much risk of fiber fracture.
If we fabricate a FLVD by the tapering method, during the tapering process, the tension could be comparatively sensitive to the variation in tapering speed, because the value of D/d is too small.
2.2 The influence of large variation in fiber diameter on the geometry of FLVD
In our previous research  it was discovered that the deformation of microstructure during the fiber drawing process can be expressed by the deformation factor (ξ):Eq. (1) and Eq. (4), if the diameters of preform and fiber, and the feeding speed of preform are given, the viscosity is linear proportional to the tension. So usually we adjust the temperature to use the tension as large as possible to draw fiber.
To give a thorough discussion, firstly we explain that, given a certain preform, if we draw it into a conventional fiber with a constant diameter, how ξ depends on d. If we use the largest tension to draw the fiber, the ξ of drawing it into a fiber with a small diameter (ξ1), is larger than the ξ of drawing it into a fiber with a large diameter (ξ2), because the viscosity of the former is less than the viscosity of the latter. It can be deduced according to Eqs. (1) and (4):
In Eq. (8) the subscript 1 denotes the fiber with a small diameter and subscript 2 denotes the fiber with a large diameter. F1 is less than F2, because the largest tension the fiber with a small diameter can withstand, is less than that the fiber with a large diameter can withstand.
During the fiber drawing process of FLVD, the fiber drawing speed is a variable. The tension increases with the increasing fiber drawing speed. The diameter of FLVD decreases with the increasing fiber drawing speed. Therefore, the smallest diameter of FLVD corresponds to the largest fiber drawing tension. So the largest tension which can be used for the fiber drawing of FLVD, is the tension that the fiber segment with the smallest diameter can endure. It means that when drawing other segments with larger diameters, the tension used is lower than the highest tension these segments can withstand. Suppose that the smallest diameter is d1 (in Eq. (8)) and the distortion factor of this segment is ξa, ξa = ξ1. Suppose that the largest diameter is d2 (in Eq. (8)) and the distortion factor of this segment is ξb, ξb>ξ2. According to Eq. (7) the ξ shows no dependence on the fiber drawing speed. Namely ξ is independent on the fiber diameter, so ξa = ξb = ξ1>ξ2. In a word, the FLVD will have the same distortion factor in the whole fiber length, no matter how much the diameter varies. The distortion factor of the microstructure is determined by the fiber segment with the smallest diameter. Since the reference of the distortion factors is the geometry of preform, if the distortion factors are the same for all segments of FLVD, the geometries are the same for all segments of FLVD.
If the highest tension which a fiber can withstand is proportional to the area of the fiber cross section, from Eq. (8) we can know that the viscosity is proportional to the square of fiber diameter. Therefore if there is a notable difference in diameter, η2 can be much larger than η1. Consequently ξa or ξb can be quite larger than ξ2. Compared with the microstructural geometry of the conventional fiber with a constant diameter of d2, the microstructural geometry of the FLVD can be degraded heavily.
Since the segment with the smallest diameter of FLVD can have a very small and fragile microstructure, in many cases it is in nanoscale, the geometry deformation of FLVD can be large in these cases. From Eq. (7) we know ξ is very sensitive to the hole size in the preform. Therefore, by using a preform with large holes, ξ can be restrained effectively. The deformation is proportional to the surface tension. For some materials, for example polymer, it is convenient to be drawn into FLVD. For other materials, for example silica glass, which can withstand high tension during the fiber drawing process, it is also convenient to be drawn into FLVD. Additionally, the preform of hollow-core photonic bandgap fiber, which is characterized with large and uniform holes, must be suitable to be fabricated into FLVD.
3. FLVD demonstration
The composition of the tellurite glass, the detailed preform preparation and fiber drawing processes can be found in . A spool of the fabricated FLVD is shown in Fig. 2 . In this work, the feeding speed of preform is 0.24 mm/min. The smallest diameter of the FLVD is 80 μm. The largest tension which the segment with the smallest diameter can withstand is around 45 g. The largest diameter of the FLVD is 320 μm. The slowest fiber drawing speed is 0.2 m/min. The fast is 3.2 m/min. The diameter of the FLVD can vary from 320 μm to 80 μm in the fiber length around 10 cm. Based on the values of F, Vf, and Vp, we drew a diagram which indicates how ρ depends on N. The results are shown in Fig. 3 . The blue curve was obtained by: . Here ρ was calculated according to the variation in Vf instead of d, because d was measured at the point far from the neck region, and the variation in measured d could not correspond to the variation in tension timely. The red dots were obtained by: .
The cross sections of the preforms and FLVDs are shown in Fig. 4 . Firstly we used preform a to draw the FLVD, the deformation was severe. Then we used preform d to draw the FLVD. The deformation was alleviated greatly. It should be explained that for some glass materials which have lower surface tension, or which can withstand higher tension, it is not necessary to use a preform with so large holes as those in inset d. If the smallest diameter of the FLVD is not reduced into such a small scale, a preform with so large holes is also not necessary. Additionally, from Fig. 4 we can see that the geometries of FLVDs keep constant no matter what the diameters are. It verifies the analyses in 2.2.
The optical loss for the FLVD was measured by using the standard cutback measurement technique. To ensure the light was coupled into the core, firstly we observed SC from the FLVD by using a 1064 nm picosecond fiber laser (the measurement is introduced in the section of 4). After then a CW laser replaces the picosecond fiber laser for the loss measurement. The measured FLVD contains a tapered segment where the diameter is reduced from 320 μm to 80 μm. The loss is around 5 dB/m at 1064 nm. We also measured the loss of a segment which has a constant diameter of 320 μm. The loss is also around 5 dB/m. It means that the tapered segment does not induce a significant extra loss. In reference  it is indicated that, for the fundamental mode of the tapered fiber, when the angle of taper slope is less than the critical slope angle, the loss induced by taper is insignificant. The critical slope angle can be calculated by:
In Eq. (9) θc is the critical slope angle. n1 and n2 are the refractive indices of cladding and core, respectively. neff is the effective index of the fundamental mode. For our FLVD though it is not a single mode fiber (SMF), in practical applications we can selectively pump the fundamental mode and restrain the high order modes by the techniques such as bending the fiber to a certain extent. In this case the main power of light is propagated in the form of fundamental mode. The calculated θc is 40°. The θc is large. It is mainly due to the small ratio of n1 to n2. Suppose that the diameter decreases linearly, for a 10 cm long FLVD with the largest diameter of 320 μm and the smallest diameter of 80 μm, the angle of taper slope is only 0.14°. Therefore the loss induced by the varying diameter is insignificant.
4. SC generation
One segment of the FLVD shown in inset e and f of Fig. 4 was used for SC generation. The core diameters along the fiber length and the dispersion curves are shown in Fig. 5 . The dispersions were calculated by the fully vectorial finite difference method. For the 0.8 μm core diameter, the zero dispersion wavelength (ZDW) is 935 nm. For the 3.2 μm core diameter, the ZDW is 1505 nm. With the decreasing core diameter, the ZDW is shifted to short wavelengths. Additionally, with the decreasing core diameter, the nonlinear coefficient increases. The calculated nonlinear coefficient for the 3.2 μm core diameter is 760 W−1km−1, and for the 0.8 μm core diameter is 7184 W−1km−1.
The pump laser was a 1064 nm picosecond fiber laser. The pulse width was 15 ps. The repetition rate was 80 MHz. The laser was connected with a silica SMF by a connector. Beam from the SMF was collimated into parallel by a lens of 0.25 NA. The parallel beam was focused and coupled into the FLVD by a lens of 0.4 NA. The coupling efficiency, defined as the launched power divided by the incident power on the lens, was about 30%. The output end of the fiber was mechanically spliced with a silica large-mode-area-fiber by using the butt-joint method. The other end of the large-mode-area-fiber was connected to an optical spectrum analyzer.
The measured SC spectra of the FLVD are shown in Fig. 6(a) . With the highest pulse energy of 5.8 nJ, the average power is 464 mW, and the peak power is 387 W. The SC spectrum covers 600-1800 nm. The 20 dB bandwidth covers 1035-1615 nm. The visible emission with the peak wavelength at 532 nm is considered to be the SHG of 1064 nm pump pulse. The reasons are as follows: The peak wavelength is just half of the pump wavelength. The intensity is found to be proportional to the square of pump power. Additionally, the peak wavelength does not shift with the increasing pump power, so it is unlikely to be dispersive wave. If it is dispersive wave, when the solitons exhibit red shift with the increasing pump power, conventionally the dispersive waves shift correspondingly. Generally the generation of second harmonics in glasses is forbidden, because the amorphous glass materials are inversion symmetric and therefore the secondary nonlinear electric susceptibility is zero. However SHG in glass fibers has already been demonstrated in many cases before [17–19]. So far the mechanisms of SHG in glass fiber have not been thoroughly understood yet. It is thought that this phenomenon has a close relationship with the defect states in the band gap of the glasses . The peak at 1150 nm is the first order stimulated Raman scattering (SRS). According to Fig. 5 we know the FLVD is initially in normal dispersion for the pump pulse. With the decreasing diameter, the dispersion increases to zero, and then it is anomalous. Meanwhile the nonlinearity increases due to the decreasing core diameter. Here we can only observe a small peak of the first order SRS, since when the pulse reaches the tapered segment other mechanisms including self phase modulation, four wave mixing, and solitons dominate due to the enhanced nonlinearity and the varying dispersion .
For the comparisons we measured SCs by using the segments of FLVD which have the constant core diameters of 3.2 μm and 0.8 μm, respectively. The fiber lengths are 80 cm and 40 cm, respectively. The results are shown in (b) and (c) of Fig. 6. In (b) the pump pulse is in normal dispersion, we can see clearly two orders of SRSs. With the highest pulse energy the SC broadening is still quite poor. In this case SRS is the main mechanism of SC broadening. The walk-off between new frequency and pump pulse, which is due to the unflattened dispersion, restrains higher order SRS. In (c) the SC threshold is low due to the enhanced nonlinearity. However, the broadest spectrum with the highest pulse energy is much narrower than that in (a). For (a), (b) and (c), the highest pulse energies are close to the damage thresholds of the pumped facets of the fibers. Over the highest pulse energies the SC spectra are unstable, and the facets of fibers are subject to irreversible damage . In (c) no SHG was observed. The possible reason might be that the launched pulse energies were too low. Though the fiber with 0.8 μm core diameter has very high nonlinearity, the damage threshold is reduced greatly.
We have demonstrated theoretically and experimentally that during the fiber drawing process, the tension is linearly proportional to the natural logarithm of fiber drawing speed. Namely, the tension is not very sensitive to the fiber drawing speed. By using a large diameter ratio of preform to fiber, this sensitivity can be decreased. Owing to the low sensitivity we can realize the FLVD with diameter varying sharply in a short fiber length by changing the fiber drawing speed dramatically.
We have also shown that the microstructural geometry of the FLVD is independent on the fiber drawing speed. The geometry can be the same for the whole fiber length. The deformation factor of FLVD is determined by the segment with the smallest diameter.
By using a preform with large holes, we have realized a FLVD of which the diameter decreases by 75% in a fiber length of 10 cm. By using this fiber we demonstrated the 600-1800 nm SC generation and the 532 nm SHG pumped by a picosecond fiber laser. The SC spectra by the conventional fibers with the largest and the smallest diameters of the FLVD were also shown, respectively. The comparisons indicated that the FLVD had the broadest SC spectrum due to its high nonlinearity, varying dispersion, and high damage threshold.
This work paves the way for the realization of various novel FLVDs efficiently and cost effectively for nonlinear optical applications.
Meisong Liao acknowledges the support of the JSPS Postdoctoral Fellowship. The authors acknowledge the support of MEXT, the Support Program for Forming Strategic Research Infrastructure (2011-2015).
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