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We analyzed the amplification effect of orthogonally polarized pulse trapping in birefringent fibers both experimentally and numerically. Trapped pulses were amplified over a wide wavelength range of 1650-1800 nm accompanying the red-shift. The maximum effective gain was 26 dB for a 140 m-long low birefringent fiber. It was clarified that this amplification effect is caused by stimulated Raman scattering between orthogonally polarized pulses.
©2010 Optical Society of America
1. Introduction
In this decade, the field of nonlinear fiber optics has developed rapidly, especially owing to the progress of photonic crystal fibers, highly nonlinear fibers, and ultrashort pulse sources [1–3]. Using a combination of an ultrashort-pulse fiber laser and a highly nonlinear fiber, light sources of wavelength-tunable ultrashort pulses and ultrawideband supercontinua have been demonstrated [3–8]. These light sources are useful for optical communication, spectroscopy, etc. Using a supercontinuum, we can also measure the carrier envelope offset frequency for an optical frequency comb. Ultrashort pulse fiber lasers are the most promising candidate for practical optical frequency comb sources [9].
When optical pulses collide in a fiber, an interaction between the pulses is induced through nonlinear optical effects. In 1989, Islam et al. discovered the phenomenon of soliton trapping in low birefringent optical fibers, in which two orthogonally polarized equal-intensity soliton pulses trap each other and co-propagate along the fiber [10]. This phenomenon is observed at a low power level at which the effect of Raman scattering is negligible.
In 2002, Nishizawa and Goto discovered a new phenomenon, pulse trapping by a femtosecond soliton pulse across the zero-dispersion wavelength (λ0) [11]. In this phenomenon, an optical pulse in the normal dispersion region is trapped by a femtosecond soliton pulse in the anomalous dispersion region, and the trapped pulse and soliton pulse co-propagate along the fiber. As the fiber input power of the soliton pulse is increased, the wavelength of the soliton pulse is shifted toward the longer wavelength side due to the soliton self-frequency shift (SSFS), and that of the trapped pulse is shifted toward the shorter wavelength side. Since then, these phenomena have been analyzed both experimentally and theoretically [12–17]. Ultrafast all-optical switching at ~1 THz has been demonstrated using this pulse trapping phenomenon [18]. A new theoretical model has been developed by Gorbach and Skyrabin [13]. Recently, continuous wavelength tuning of ultrashort pulses in the visible region has been demonstrated using pulse trapping phenomena in PCF [19].
In 2003, Nishizawa and Goto also discovered another pulse trapping phenomenon in birefringent fibers: trapped pulse generation (amplification) [20]. When orthogonally polarized pulses are temporally overlapped in polarization maintaining fiber (PMF), if the group velocity matching condition is satisfied, they trap each other through cross-phase modulation (XPM) and co-propagate along the fiber. When the wavelength of an optical pulse is shifted toward longer wavelength side by SSFS, an orthogonally polarized pulse is also red-shifted through XPM so that the group velocity matching condition is always maintained. During the propagation, the optical pulse at the longer wavelength side suffers the Raman gain of the orthogonally polarized pulse at the shorter wavelength, and the pulse energy of the ultrashort pulse at the shorter wavelength side is gradually transferred to the trapped pulse at the longer wavelength. Thus, we can amplify the trapped pulse using this orthogonally polarized pulse trapping phenomenon. Ultrafast all-optical switching has also been demonstrated using this phenomenon [21]. The amplification characteristics, however, have not been investigated yet.
In this paper, we investigated the amplification effect of pulse trapping in PMF both experimentally and numerically. From our analyses for several kinds of PMF, it was clarified that the amplification effect is caused by the orthogonally polarized Raman gain. Using this effect under the group velocity matching condition, the amplification of trapped pulses was demonstrated over a wide wavelength range of 1650-1800 nm. The obtained maximum effective gain was 26 dB in a 140 m-long low birefringent fiber.
2. Experimental and numerical analysis methods
2.1 Experimental setup
Figure 1 shows the experimental setup used for pulse trapping in birefringent fibers. The phenomenon of pulse trapping was efficiently induced by adjusting the incidence conditions of two pulses [21]: the two pulses were orthogonally polarized and completely temporally overlapped, and the wavelength relation between the two pulses satisfied the group velocity matching condition. To achieve these pulse conditions, we used the same orthogonally polarized, widely wavelength-tunable, two-color femtosecond soliton pulse source as reported in other studies [3,4,21]. As the pump pulse for this soliton pulse source, a high-energy Er-doped fiber laser (IMRA, μJewel) was used. It generated ~1 ps pulses at a wavelength of 1560 nm. The repetition frequency was 200 kHz. The pump pulses were coupled into a wavelength conversion PMF (WC-PMF, Fujikura, 17 m) after adjusting their input energy. The polarization direction of the pump pulses was inclined from the birefringent axis to generate two orthogonally polarized soliton pulses. The wavelength tuning range of the generated pulses was 1560-1800 nm. The temporal widths of the pulses were 150-190 fs full width at half maximum (FWHM) assuming a transform-limited sech2 pulse shape. The temporal difference between the two pulses was compensated for using delay lines consisting of a PBS and corner mirrors. Then, the two pulses were coupled into the sample PMF after aligning their polarization directions to the birefringent axes: the polarization directions of the control pulse at the shorter wavelength and the signal pulse at the longer wavelength were aligned along the slow and the fast axes, respectively. The energies and spectra of the output pulses from the sample PMF were measured after filtering either the control or signal pulses using a polarizer.
Fig. 1 Experimental setup for pulse trapping. HWP, half wave plate; WC-PMF, wavelength conversion PMF; PBS, polarizing beam splitter; DL, delay line.
2.2 Numerical analysis
We also analyzed the phenomenon of pulse trapping numerically. The electrical field E(z,t) of two orthogonally polarized pulses is given by [1,22]
where z is distance, t is time, and indicate the directions along the slow and fast axes of the PMF, A and B are the envelopes of the control and signal pulses which propagate along the slow and fast axes, and β0A and β0B are the propagation constants of the fast and slow axes at the optical frequencies ω0A and ω0B, respectively. Evolutions of the pulses (A and B in Eq. (1)) along a birefringent fiber are represented by the coupled nonlinear Schrödinger equations [1,21,23], where T = t - β1A(ω0A)z, and β1A is the first-order dispersion of the slow axis. The left sides in the above equations represent linear effects, including the effects of chromatic dispersion, loss, and birefringence, where β2j and β3j with j = A or B are the second- and third-order dispersion parameters, αj is the attenuation constant, and d is the walk-off parameter representing the birefringence. When the group velocity matching condition is satisfied, the walk-off parameter d = β1A(ω0A) - β1B(ω0B) = 0. The magnitudes of the dispersion parameters were measured using the wavelength-tunable soliton pulses [24]. On the other hand, the right sides in Eqs. (2) and (3) represent nonlinear effects, including self-phase modulation, cross-phase modulation, self-steepening, Raman scattering, and the effect of Raman gain, where γj with j = A or B is the nonlinear parameter, and TR is the parameter corresponding to the Raman response time. The gain coefficients gA and gB, and Raman gain coefficient gR are related bywhere Aeff is the effective core area. The magnitude of the Raman gain coefficient gR will be discussed in the following section. The coupled amplitude equations were analyzed with the split step Fourier method [1].Figure 2 shows the numerical results of evolutions of (a) spectral and (b) temporal waveforms of the control and signal pulses in the pulse trapping along a 200 m-long PMF. The wavelength of the trapped signal pulse is red-shifted by SSFS of the twin pulses in the propagation along the PMF. At the same time, the energy of the control pulse is transferred to the trapped signal pulse. Consequently, the trapped signal pulse is gradually amplified along the fiber. Throughout the propagation, the waveform of the trapped pulse retains a nearly sech2-shape due to the soliton effect. The temporal width (FWHM) is 200-300 fs. In addition, the wavelength relation between the two pulses satisfies the group velocity matching condition, and the two pulses are completely temporally overlapped.
Fig. 2 Numerical results of evolutions of the control (blue line) and the signal pulses (red line) in the propagation along a 200 m PMF: (a) spectra (Media 1), (b) temporal waveforms (Media 2). The input energies of the control and the signal pulses are 250 pJ and 4 pJ, respectively. The group velocity matching condition is satisfied. The zero position of the time scales is always adjusted to the peak point of the control pulse.
3. Gain coefficient of pulse trapping
First, we examined the mechanism of the amplification effect for the orthogonally polarized pulse trapping. Strictly speaking, the trapping efficiency is not perfect. For accurate investigation of the gain coefficient, we introduced the concept of the energy of effective signal pulse, which is a trapped part of the input signal pulse. Figure 3 shows representative output spectra. The trapped signal pulse is red-shifted and amplified by the control pulse, and has a sech2-shape. The non-trapped part of the input signal pulse (non-trapped signal pulse) is stranded at the initial wavelength. Meanwhile, when only the signal pulse is present, the spectrum is not changed from the original form in the propagation because of the small fiber loss and negligible nonlinear effect. We call this unchanged output signal pulse as original signal pulse. The effective signal pulse corresponds to the difference component between the original and the non-trapped signal pulses. The energy of the effective signal pulse Eeff is calculated by Eeff = Es - En, where Es and En are the energy of the original signal pulse and the non-trapped signal pulse, respectively. In other words, the effective signal pulse is trapped by the control pulse at the initial stage of the propagation along the PMF, and then observed as the trapped signal pulse after the propagation. The effective gain is defined by Geff = 10log(Et/Eeff), where Et is the energy of the trapped signal pulse. The trapping efficiency is given by R = Eeff/Es. In this work, the trapping efficiency R was found to be 0.1-0.6 for the experimental analyses, and 0.80-0.99 for the numerical ones. It is considered that the degradation of the trapping efficiency for the experimental analyses is caused by the timing jitter and amplitude fluctuation of the pulse source. However, the effective gain Geff is unaffected by the difference of the trapping efficiency R between both analyses.
Fig. 3 Representative output spectra from the fast axis of PMF. The solid line shows the trapped and the non-trapped signal pulse when the control and signal pulse is coupled into the PMF. The broken line shows the signal pulse when only the signal pulse is present, which corresponds to the original signal pulse. The dotted line on the trapped signal pulse is the sech2 fit. Inset shows a magnified view. The difference component between the original signal and the non-trapped signal pulses corresponds to the effective signal pulse.
The effective gains Geff were analyzed for three kinds of PMFs having different magnitudes of birefringence ranging from 2.7 × 10−4 to 5.0 × 10−4: PMF1, a standard low birefringent fiber (3M, FS-CG-7421); PMF2, a standard highly birefringent fiber (3M, FS-PM-7811); and PMF3, a highly nonlinear and highly birefringent fiber (Furukawa). The parameters of these PMFs are shown in Table 1 . Table 2 shows some conditions of the input pulses. The wavelength relations satisfy the group velocity matching condition, and the corresponding frequency differences are 4.6-14.0 THz.
Table 1. Parameters of sample PMFs at a wavelength of 1550 nm
Table 2. Conditions of the input pulses
Figure 4(a) shows the effective gains Geff versus the energy of the effective signal pulse Eeff for the three kinds of PMF. As the energies Eeff are increased, the gains Geff are continuously decreased due to the output saturation. Conversely, the gains Geff are increased and gradually saturated with decreasing energy Eeff. A maximum effective gain of 26 dB was achieved by using PMF1. As shown in Fig. 4 (a), the numerical results are fitted with the experimental ones by substituting suitable Raman gain coefficients gR into Eq. (4): for PMF1, gR = 2.91 × 10−15 m/W; for PMF2, gR = 2.18 × 10−15 m/W; and for PMF3, gR = 1.97 × 10−15 m/W. The magnitudes of the Raman gain coefficients gR are plotted together with the Raman gain profile in Fig. 4(b). From Fig. 4(b), we can say that the amplification effect in the pulse trapping along the PMF is caused by the orthogonally polarized Raman gain g ⊥. That is valid because the polarization directions of the control and the signal pulses are orthogonally polarized relative to each other throughout the propagation along the PMF. The orthogonally polarized Raman gain coefficient g ⊥ is smaller than the co-polarized one g //. However, as shown in Fig. 2, the control pulse is an ultrashort pulse with a high peak power on the order of kilowatts, and it is overlapped with the signal pulse throughout the propagation in the fiber. Thus, SRS is always active in the propagation. As a result, a large gain is achieved in a comparatively short PMF of only about 100 m, such as 26 dB in PMF1. Furthermore, the amount of energy transferred from the control pulse to the trapped signal pulse is increased in propagation through 140-200 m of PMF. It is expected that the maximum gain will increase if a longer PMF is used.
Fig. 4 (a) Effective gain Geff as a function of energy of effective signal pulses in PMF1-3. The filled symbols and the dashed lines are the experimental and the numerical results, respectively: circles, PMF1; squares, PMF2; triangles, PMF3. (b) Raman gain spectra for a silica fiber [25] at a pump wavelength of 1.65 μm when two pulses (pump and Stokes pulses) are co-polarized (g //) and orthogonally polarized (g ⊥). The filled circular, square, and triangular symbols are the obtained Raman gain coefficients used in the numerical analyses for PMF1-3, respectively.
4. Wideband amplification
Next, we examined the amplification effect of the pulse trapping over a wide wavelength range in a standard low birefringent 140 m-long PMF (also used in Section 3 as PMF1). Figure 5(a) shows the relative delay time versus wavelength, which was measured using the wavelength tunable soliton pulses [24]. The delay times (related to the group velocities) were continuously and widely matched between the control and the signal pulses by appropriately selecting their wavelengths and polarization directions. For the measurements, we chose four wavelength relations satisfying the group velocity matching conditions: the wavelengths of the control and the signal pulses were (i) 1602 and 1650 nm, (ii) 1654 and 1700 nm, (iii) 1711 and 1750 nm, and (iv) 1766 and 1800 nm. The corresponding frequency differences were 5.4-3.2 THz.
Fig. 5 (a) Relative delay time of PMF1 as a function of wavelength. Wavelength relations (i)-(iv), which are used for the experimental analyses, satisfy the group velocity (the group delay) matching conditions. (b) Wideband amplification characteristics of pulse trapping; the symbols show the input-output gain GIO for small signal pulses under the wavelength relations (i)-(iv). The energies of control pulses are 205-250 pJ at the output. The dashed lines show the numerical results for a variety of input energies of the control pulses, Eic.
The amplification effect of the pulse trapping was induced under the wavelength relations (i)-(iv). To simplify the calculation of the gains, the input-output gain is given by GIO = 10 × log(Et/Es), where Et and Es are the energy of the trapped and the original signal pulses, respectively. Figure 5(b) shows the input-output gains GIO for small signal pulses (Es ~1 pJ). The energies of control pulses coupled into the PMF were adjusted to give almost constant energies of 205-250 pJ at the output under the wavelength relations (i)-(iv). The obtained gains GIO are 15-20 dB over a wide wavelength range of 1650-1800 nm. The spectral forms are nearly sech2-shaped accompanying the red-shift of 30-70 nm, as in Fig. 3. Assuming transform-limited sech2 pulses, the temporal widths (FWHM) are 220-380 fs. The frequency differences are changed from 5.4 to 3.2 THz with the incident wavelength relations (i)-(iv) and the red-shifts of the two pulses in the propagation along PMF1. However, the orthogonally polarized Raman gain coefficient g ⊥(Ω) shown in Fig. 4(b) is comparatively flat over the frequency difference range. Even if we consider that the Raman gain coefficient is inversely proportional to wavelength, it leads to only a small change of g ⊥(Ω) from 2.96 × 10−15 to 2.76 × 10−15 m/W. Besides, the fiber loss is low, at <1.5 dB, for a 140 m-long fiber at a wavelength of 1800 nm. As a result, the trapped signal pulses are amplified over a wide wavelength range of 150 nm.
The pulse trapping, or the soliton effect, is induced in the anomalous dispersion region of the fibers. At the longer wavelength side, the amplification is limited by the higher fiber loss caused by infrared absorption. Conversely, at the shorter wavelength side, the signal pulses of >1500 nm are trapped by the control pulses of >1400 nm in a standard PMF. According to the numerical results shown in Fig. 5(b), the gain is obtained over a wide wavelength range of 350 nm (1500-1850 nm) for PMF1 by coupling the input energy of the control pulse of 400 pJ. It should be possible to achieve more wideband amplification, for example, at wavelengths >800 nm, by using dispersion managed photonic crystal fibers.
5. Conclusion
In this paper, we analyzed the amplification characteristics of orthogonally polarized pulse trapping by using several kinds of PMF both experimentally and numerically. When the signal pulse is trapped by the control pulse and they co-propagate along the PMF, the trapped signal pulse is amplified by the control pulse. It was clarified that this amplification effect of pulse trapping is caused by the orthogonally polarized Raman gain. The observed maximum effective gain was 26 dB using only a 140 m-long PMF. The amplification was demonstrated over a wide wavelength range of 150 nm in the anomalous dispersion region. It is expect that the wavelength range for amplification can be as high as 350 nm or more. These results suggest that we can demonstrate ultrafast all-optical control techniques including a wideband amplifier using ultrashort pulse source and a PMF of about a hundred meters long.
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