We investigated the phenomenon of orthogonally polarized pulse trapping between a continuous-wave beam and an ultrashort soliton pulse in birefringent fibers both experimentally and numerically. Using pulse trapping and amplification, we demonstrated ultrashort pulse generation from a continuous-wave beam. The generated pulse had a nearly transform-limited sech2-shape and a temporal width of 350 fs. The obtained maximum pulse energy was 300 pJ using a 400 m-long low-birefringence fiber, and the corresponding gain was as large as 41 dB.
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Ultrafast nonlinear optical effects in optical fibers are useful for a variety of techniques, such as mode-locking , supercontinuum generation [2–6], wavelength-tuning [6–10], and all-optical control [1, 6, 10–12]. These are very important for spectroscopy, nonlinear microscopy, metrology, ultrafast and wideband optical fiber communication, all-optical signal processing, etc. We can obtain a variety of nonlinear effects effectively by coupling ultrashort light pulses in optical fibers.
In the anomalous dispersion region of the fiber, an ultrashort soliton pulse is generated in balance between the linear effect of group velocity dispersion (GVD) and the nonlinear effect of self-phase modulation . When the pulse energy is increased, the optical frequency of the pulse is downshifted (the wavelength is shifted toward the longer wavelength side) due to a soliton self-frequency shift (SSFS) originating from intra-pulse stimulated Raman scattering . Widely wavelength tunable pulse sources have been reported by applying such red-shifted ultrashort soliton pulses [6–9].
When another optical pulse collides with the ultrashort soliton pulse in optical fibers, the beams interact through nonlinear optical effects such as cross-phase modulation (XPM) and stimulated Raman scattering (SRS). In particular, when an optical pulse in the normal dispersion region satisfies the group velocity matching condition with the soliton pulse in the anomalous dispersion region across the zero-dispersion wavelength (λ 0), the pulse trapping phenomenon occurs . In the propagation along the fiber, when the soliton pulse is red-shifted by SSFS, the wavelength of another pulse in the normal dispersion region is shifted toward the shorter wavelength side (blue-shift) through XPM to maintain the group velocity matching condition. Thus, the soliton pulse traps another pulse, and the two pulses co-propagate along the fiber. The characteristics of this phenomenon have been analyzed by many groups [15–20]. A new theoretical model of gravity-like potentials has been developed by Gorbach and Skyrabin . Using this pulse trapping across λ 0, ultrafast all-optical switching [6, 11] and continuous wavelength tuning of visible ultrashort pulses  have been demonstrated.
Two orthogonally polarized equal-intensity soliton pulses colliding in a birefringent optical fiber induce the soliton trapping phenomenon, which was discovered by Islam in 1989 . The two soliton pulses trap each other and co-propagate along the fiber. This phenomenon is observed for pulses with low power and a duration on the order of picoseconds, conditions at which the effect of Raman scattering is negligible.
Furthermore, when two orthogonally polarized ultrashort pulses are aligned along the slow and fast axes of a birefringent fiber, and they have appropriately shorter and longer wavelengths λS and λL at the anomalous dispersion region, respectively, the group velocity matching condition, vg_fast(λL) = vg_slow(λS), is satisfied. When these two pulses are temporally overlapped in a birefringent fiber, they undergo the pulse trapping phenomenon known as trapped pulse generation (amplification) . In the propagation along the fiber, the pulses maintain ultrashort sech2-shape waveforms by the soliton effect. And, when an ultrashort pulse is red-shifted due to SSFS, the orthogonally polarized pulse is also red-shifted through XPM so that the group velocity matching condition is always maintained. Thus, the two pulses trap each other and co-propagate along the fiber. During the propagation, since the ultrashort pulse at the longer wavelength side experiences the Raman gain of the orthogonally polarized pulse at the shorter wavelength side, the pulse energy is gradually transferred from the shorter-wavelength pulse to the longer-wavelength pulse. Consequently, we can amplify the optical pulse by a large amount through the Raman gain of the orthogonally polarized pulse. Using this phenomenon of orthogonally polarized pulse trapping and amplification, ultrafast all-optical switching [6, 12] and ultrashort pulse amplification of ~26 dB  have been demonstrated.
So far, orthogonally polarized pulse trapping has been investigated for pairs of pulses. Since an ultrashort pulse with high peak power acts as the pump pulse, we can obtain a large gain for pulse trapping and amplification by using only a few hundred meters of PMF. Using the pulse trapping and amplification, we can generate an ultrashort pulse and temporally overlapped twin ultrashort pulses. For the pulse trapping of a pulse pair, the temporal relation has to be adjusted accurately. If we can demonstrate the pulse trapping of a continuous-wave (cw) beam, ultrashort twin pulses can be generated stably. So far, pulse trapping of a cw beam has not been investigated yet.
In this paper, we report the first demonstration of the phenomenon of orthogonally polarized pulse trapping between a cw beam and an ultrashort pulse in a low-birefringence polarization maintaining fiber (LB-PMF). We demonstrated and investigated the ultrashort pulse generation from a cw beam using pulse trapping and amplification by an ultrashort pump pulse both experimentally and numerically. In a few meters of propagation along the LB-PMF, a part of the cw beam was trapped, and an ultrashort pulse was generated by the pump pulse. Then, the generated pulse was amplified by the pump pulse, and a large gain of 41 dB was obtained using a 400 m-long LB-PMF. The generated pulse had a nearly transform-limited sech2-shaped ultrashort pulse waveform.
2. Methods for numerical and experimental analysis
2.1 Numerical analysis
We analyzed the phenomenon of orthogonally polarized pulse trapping between a cw beam and an ultrashort pulse numerically. Evolutions of the two optical beams along the PMF are represented by the coupled nonlinear Schrödinger equations . The coupled amplitude equations were analyzed with the split step Fourier method . For the analysis of the behavior, we produced spectrograms from the achieved electric field envelopes of propagating pulses, Asig, using the equation for polarization-gate cross-correlation frequency resolved optical gating (PG-XFROG) ,
2.2 Experimental setup
Figure 1 shows the experimental setup used for the ultrashort pulse generation from a cw beam using orthogonally polarized pulse trapping and amplification. As an ultrashort pump pulse source, a passively mode-locked Er-doped fiber laser (IMRA, Femotolite780) was used. The pump pulse was amplified by an Er-doped fiber amplifier. The temporal width of the ultrashort pump pulse was 730 fs full width at half maximum (FWHM) and the center wavelength was 1556 nm. The repetition frequency was 50 MHz. On the other hand, as a cw beam source, a wavelength tunable laser diode (Santec, TSL-510) was used. The spectra of the two beams are shown in Fig. 4(a) . The temporal waveform of the pump pulse, which was observed using the second-harmonic generation FROG (SHG-FROG) technique, is shown together. These beams were combined using a polarizing beam splitter. Then, an ultrashort pulse was generated from the cw beam by coupling the two beams into the LB-PMF. The mode field diameter of the LB-PMF was 10.5 μm, the magnitude of birefringence was 2.7 × 10−4, and the second- and third-order dispersion parameters β2 and β3 were −22 ps2/km and 0.12 ps3/km, respectively. The fiber lengths were 100, 300, and 400 m. To satisfy the group velocity matching condition between the two beams, the wavelength of the cw beam was adjusted to 1604 nm, and the polarization directions of the pump pulse (at the shorter wavelength side) and the cw beam (at the longer wavelength side) were aligned along the slow and the fast axes of the LB-PMF, respectively. At the fiber output, the powers, spectra, and auto-correlation traces were observed.
3. Demonstration and characterization of ultrashort pulse generation from cw beam
3.1 Analysis of mechanism
First, we analyze the mechanism of the ultrashort pulse generation from a cw beam, numerically. The fiber parameters are set to the same conditions as those of the LB-PMF used in the experiment. The nonlinear coefficient γ and the Raman gain coefficient gR are 1.4 W−1km−1 and 3 × 10−15 m/W, respectively. To simplify the analysis, a transform-limited (TL) sech2-shaped ultrashort pulse (560 pJ, 300 fs, 1556 nm) is assumed as an input pump pulse. For a cw beam, a sufficiently long pulse with temporal width of 1 us and peak power of 13 mW at wavelength of 1604 nm is assumed. Figure 2 shows the numerical results for the evolutions of the cw beam and ultrashort pulse in the propagation along the PMF up to an initial point at L = 16 m. The spectrograms shown in Fig. 2(a) and (b) are obtained using Eq. (1), in which the reference pulse (absolute value of Aref) has a TL sech2-shape and a temporal width of 500 fs (FWHM). Figure 2(c) and (d) are the corresponding temporal waveforms and spectra, respectively. In the propagation along the PMF, the pump pulse along the slow axis becomes a soliton pulse by the soliton effect. The wavelength of the soliton pulse is shifted toward the longer wavelength side due to SSFS, and the pulse is delayed because of GVD. The energy of the soliton pulse is 480 pJ. The conversion efficiency from the pump pulse to the soliton pulse is 86%. Because this soliton pulse effectively works as a pump for the Raman gain later, we call this converted pulse the pump soliton pulse. On the other hand, in the initial few meters of propagation along the PMF, another pulse along the fast axis is generated from the cw beam at the same delay time as the pump soliton pulse. We call the new generated pulse from the cw beam the generated pulse.
The mechanism of cw trapping is explained as follows. The temporally overlapped part of the cw beam experiences XPM due to the pump pulse. As a result, the part overlapping the leading edge of the pump pulse is red-shifted, and the part overlapping the trailing edge of the pump pulse is blue-shifted by XPM. Owing to the anomalous dispersion, the modulated spectra gather, gradually forming the pulse. Since the generated pulse is temporally overlapped with the pump soliton pulse, the generated pulse is trapped by the pump soliton pulse through XPM, and they co-propagate along the fiber. In addition, the energy of the pump soliton pulse is transferred to the generated pulse through orthogonally polarized Raman gain. These behaviors are the same as the phenomenon of orthogonally polarized pulse trapping and amplification that was investigated by use of twin pulses in earlier studies [6, 12, 22, 23].
Here, we introduce a trapping efficiency for the overlapped component. In this case, it is assumed that a 300 fs component of the cw beam is overlapped with the pump pulse at the fiber input due to the temporal width of the input pump pulse. The energy of the overlapped cw beam, which is estimated from the temporal width multiplied by the power of the cw beam (300 fs × 13 mW), is 3.9 fJ. Because the energy of the trapped component of the cw beam is 3.3 fJ, which is estimated from the dip component of the cw beam observed as a result of trapping, such as that shown at the −1 ps point of L = 10 m in Fig. 2(c), the trapping efficiency for the overlapped component is calculated to be 85%. Thus, most of the cw beam overlapped with the pump pulse at the fiber input is trapped by the pump pulse. The red-shifted pump soliton pulse does not trap the other part of the cw beam because of the group velocity mismatching.
Also, after the pulse generation, the pump soliton pulse and generated pulse co-propagate along a long length of the LB-PMF by orthogonally polarized pulse trapping. Because of the soliton effect and SSFS by the two pulses, the pump soliton pulse maintains the ultrashort sech2-shape accompanying the red-shift. On the other hand, the generated pulse gradually forms an ultrashort sech2-shape accompanying the red-shift. In addition, the generated pulse is largely amplified by the pump soliton pulse owing to accumulation of the energy exchange by the orthogonally polarized Raman gain . Figure 3 shows the numerical results of spectral and temporal waveforms at the L = 300 m point in the LB-PMF. The generated pulse has a large energy of 123 pJ. The waveform is a nearly TL sech2-shaped ultrashort pulse. Moreover, the generated pulse and the pump soliton pulse are completely temporally overlapped.
3.2 Demonstration of ultrashort pulse generation from cw beam
Figure 4 shows the experimental results of spectra for the pulse generation from the cw beam using the 300 m-long LB-PMF. Figure 4(a) shows the spectra of the input ultrashort pump pulse (at 1556 nm) and the cw beam (at 1604 nm). Figure 4(b) shows the spectra at the fiber output when only the 1080 pJ pump pulse was coupled into the fiber. A 470 pJ soliton pulse was observed at 1698 nm in addition to the remainder of the pump pulse. The conversion efficiency from the input pump pulse to the soliton pulse was 44% because of the negative chirp of the input pump pulse shown in the inset of Fig. 4(a). Due to the low fiber loss, the energy of this soliton pulse was nearly equal to that of the pump soliton pulse at the initial stage. Next, Fig. 4(c) shows the spectra at the fiber output when a cw beam of 13 mW was coupled into the fiber with the same pump pulse. We observed another generated spectral component at 1728 nm on the longer wavelength side, which satisfied the group velocity matching condition with the pump soliton pulse. This is the spectral component generated from the cw beam through orthogonally polarized pulse trapping and amplification by the pump soliton pulse, as indicated in the numerical analysis (Section 3.1). The spectrum was fitted to a sech2-shape. The frequency bandwidth, Δf, was 0.93 THz. Figure 5(a) shows the auto-correlation trace of only the generated pulse; the pump and cw beams were removed using a low pass filter (λc = 1600 nm) and a polarizer. The temporal width (FWHM), Δt, of the generated pulse was calculated to be 350 fs from the width (FWHM) of the auto-correlation trace, ΔTgen = 540 fs, under the assumption of a sech2-shaped pulse. The time·frequency bandwidth product, Δt·Δf, was calculated to be 0.324, which was almost equal to the value of the TL sech2-shaped pulse, 0.315. Thus, the generated component was a nearly TL sech2-shaped ultrashort pulse.
We also measured the auto-correlation trace of the generated twin pulses (generated pulse and pump soliton pulse) before splitting them using the polarizer. We can achieve a narrower temporal waveform with a few peaks by the temporal coherent overlapping of two colored twin pulses compared with using one pulse. Figure 5(b) shows the auto-correlation trace of the twin pulses. The width at the inside peak, ΔTtwin = 190 fs, was narrower than that of the generated pulse alone. Since the autocorrelation trace of the twin pulses was observed to be stable, we can conclude that the twin pulses co-propagated along the fiber stably. Even if the timing jitter between twin pulses is produced by some factors, such as noise and the birefringence, the timing jitter is compensated because the pulses trap each other and they are temporally overlapped again through XPM to satisfy the group velocity matching condition.
3.3 Examination of characteristics
Next, we examined the characteristics of the pulse generation from a cw beam. Figure 6 shows the energies of the generated pulse and the pump soliton pulse at the output of the 300 m-long LB-PMF as functions of the initial energy of the pump soliton pulse and the input power of the cw beam. The higher energy of the pump soliton pulse at the initial stage was generated with an almost constant conversion efficiency of 43–45% by coupling the larger ultrashort pump pulse into the fiber. As shown in Fig. 6(a), when the initial energy of the pump soliton pulse increased, the output energy of the generated pulse from the cw beam also increased due to the larger energy shift from the pump soliton pulse. This result is explained by the fact that the generated pulse was amplified by the larger pump soliton pulse that effectively worked as pump for Raman gain in orthogonally polarized pulse trapping. Figure 6(b) shows the output energies of the two pulses as a function of the input power of the cw beam for an input pump pulse energy of 1080 pJ. The corresponding initial energy of the pump soliton pulse was 470 pJ. When the input power of the cw beam increased, the output energy of the generated pulse also increased, whereas that of the pump soliton pulse decreased. That is because the larger component of the cw beam was trapped by the pump soliton pulse, leading to larger Raman gain in the propagation along the LB-PMF.
These experimental results are in good agreement with the numerical analysis, in which the input pump pulse is assumed to be a negatively chirped, sech2-shaped, ultrashort pulse. According to this numerical analysis, the energy of the trapped component increases in proportion to the input power of the cw beam with an almost constant trapping efficiency of 72%. In the experiment, the achieved maximum energy of the generated pulse was 240 pJ for the 13 mW input cw beam and 1080 pJ pump pulse. The corresponding temporal waveform, shown in Fig. 5(a), had a temporal width Δt of 350 fs. Thus, the estimated peak power is about 600 W. In addition, we obtained a large gain of 47 dB in terms of peak power along the fast axis of the PMF. It is expected that the maximum energy and gain will increase if a pump pulse of higher energy and a cw beam of higher power are used.
Figure 7 shows the variation of energies of the twin pulses and the corresponding gain characteristic for the generated pulse as a function of the fiber length. The energy of the input pump pulse was 940 pJ, and the initial energy of the pump soliton pulse was 416 pJ, giving a conversion efficiency of 44%. The power of the cw beam was 13 mW. In the experiment, the generated pulse was not observed when the 100 m-long LB-PMF was used. However, when the 300 and 400 m-long LB-PMFs were used, the generated pulses were observed. The pulse energy of the generated pulse increased, whereas that of the pump soliton pulse decreased as the fiber length increased. This is because the energy is gradually transferred from the pump soliton pulse to the generated pulse through stimulated Raman scattering. The experimental results are in good agreement with the numerical ones. The generated pulse was amplified to 300 pJ at the 400 m point. From the numerical result, by assuming a generated pulse energy of 0.024 pJ as a seed at the initial stage (L = 10 m), we obtained a large gain of 41 dB in terms of pulse energy. This gain is larger than our previous study (26 dB)  due to the higher energy of the pump pulse and the co-propagation of the two pulses along the longer fiber. Generally speaking, the observed gain is larger than that of the conventional cw pumped fiber amplifiers such as rare-earth doped fiber amplifiers and Raman amplifiers.
In this paper, we demonstrated and investigated the ultrashort pulse generation from a cw beam using the orthogonally polarized pulse trapping and amplification in birefringent fibers both experimentally and numerically for the first time. The trapped and generated pulse was amplified by the pump pulse accompanying the red-shift. The waveform became a nearly transform-limited sech2-shaped ultrashort pulse. The gain for peak power was as large as 47 dB. The observed maximum energy was 300 pJ using a 400 m-long low-birefringence fiber. The estimated gain for the pulse energy was as large as 41 dB. In addition, we obtained a perfectly temporally overlapped ultrashort pulse pair consisting of the generated pulse and the pump soliton pulse. Owing to the stable temporal overlapping, the autocorrelation trace of the twin pulses stably formed a narrower pulse with a few peaks.
This study suggests that optical beams having a variety of temporal widths, such as any optical beams from femtosecond pulses to cw, can be trapped and amplified by another orthogonally polarized ultrashort pulse. Applying this ultrafast nonlinear interaction between optical beams, it may be possible to demonstrate highly functional ultrafast all-optical control techniques including an amplifier (>40 dB gain) using only one birefringent fiber with a length of a few hundred meters.
Part of this work was supported by the KDDI Foundation.
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