The optical solitons are intense light pulses that propagate in a nonlinear medium in an anomalous dispersion regime. They retain shape due to the balance between chromatic dispersion and self-phase modulation. The possibility of transmission of such undistorted pulses in optical fibers was predicted by Hasegawa and Tappert in 1973 [1]. In 1986, Mitschke and Mollenauer discovered that an ultrashort soliton undergoes self-frequency shift caused by energy transfer from the higher to the lower frequency part of the spectrum due to Raman scattering [2] [soliton self-frequency shift (SSFS)]. This phenomenon allows for continuous tuning of the soliton’s spectral position and facilitates reaching wavelengths not available with typical laser sources. In recent years, the possibility of effectively tuning solitons in photonic crystal fibers (PCFs) was investigated. The photonic crystal fibers appeared to be perfect media for spectral tuning of solitons as they provide control over the shape of dispersion curve and allow us to obtain a small effective mode area which results in high effective nonlinearity [3–5].

The second thread of the recent research related to solitons in optical fibers is focused on few-mode fibers (FMFs) and multimode fibers (MMFs) [6]. In reference to photonic crystal fibers, the FMFs and MMFs have relatively high effective area, thus they can guide high-intensity pulses. Moreover, they enable more complex dynamics by providing additional degrees of freedom to the system. Propagation of solitons in higher-order modes of MMF was investigated by Rishøj et al. [7]. The conversion was enabled by Raman scattering and group velocity matching of distinct spatial modes.

The Raman scattering is not the only phenomenon that is important to describe nonlinear pulse propagation. The next mechanism is the cross-phase modulation (XPM). Due to XPM, the intense pulse can force the common group velocity causing trapping of the other copropagating pulse. When two copropagating pulses are orthogonally polarized, the XPM induces nonlinear phase shifts in both polarization components which depend on the intensity of the orthogonal polarization component [8]. This kind of mutual interaction can lead to the trapping of orthogonally polarized pulses. The first studies on soliton trapping showed a numerical investigation of XPM in conventional birefringent fibers [9,10]. They were followed by the experimental demonstration of soliton trapping in low-birefringent optical fiber in the low power regime [11]. In this experiment, the linearly polarized pulse with azimuth angle $\theta = {45}^{\circ }$ excited two polarization components. The slow-axis component lowered its wavelength to increase the group velocity, and the fast-axis component raised its wavelength to decrease the group velocity. Finally, both components propagated with a common group velocity.

Interestingly, there exists a mechanism that can transfer energy from one polarization component to the other. This mechanism is provided by orthogonal Raman scattering. The energy from the shorter wavelengths in one polarization is transferred to the longer wavelengths in orthogonal polarization. Although the orthogonal Raman response is weaker than the co-polarized Raman response [12], it plays a key role in the pulse trapping observed in the conventional birefringent fiber by Nishizawa and Goto [13,14]. In those experiments, the soliton polarized along the slow fiber axis transferred part of its energy to the fast axis. The generated pulse polarized along the fast fiber axis was trapped by the slow-axis component and fed by the trapping pulse through the Raman scattering. The authors confirmed that the pulses overlapped temporally and co-propagated along the fiber. In the following work, the authors also investigated pulse trapping of co-linearly polarized pulses across the zero-dispersion wavelength [15]. Additionally, trapping of an orthogonally polarized continuous wave [16] and even pulse trapping and amplification of incoherent light from a super-luminescent diode [17] were demonstrated in birefringent fibers.

In the Refs. [13,14,16,17], the conventional birefringent fiber was used. In this type of fiber, the group velocity matching of polarization modes is possible between a shorter wavelength aligned to the slow axis and a longer wavelength aligned to the fast axis. Consequently, the pulse trapping is possible when the trapping pulse is aligned to the slow axis and the generated trapped pulse is aligned to the fast axis. Simultaneously, pumping in the fast axis is preferable in conventional birefringent fibers to avoid gradual pulse depolarization by orthogonal Raman scattering.

In birefringent PCFs, the situation is opposite. The distinguishing feature of PCFs is the negative value of group birefringence which means that the pulses polarized along the fast fiber axis have lower group velocity than pulses polarized along the slow fiber axis [18]. Consequently, a shorter wavelength aligned to the fast axis and a longer wavelength aligned to the slow axis can be group-velocity-matched. Our aim was to investigate the influence of orthogonal Raman scattering on solitons propagating in birefringent PCF. To our knowledge, this is the first detailed study of soliton trapping in optical fiber with negative group birefringence.

To observe the effect of Raman scattering on solitons, we used mode-locked Er-doped fiber laser (FemtoFiber pro IR, TOPTICA Photonics) operating at 1.56 µm and generating 23 fs pulses with a repetition rate of 80 MHz and average output power of approximately 200 mW. To control the power level and the polarization state of the laser pulses, they were passed through a polarizer and half-wave plate. For measuring an average beam power, we used a thermal power sensor (Thorlabs, S401C). The pulses were then injected into the fiber with an aspheric lens (Thorlabs, C230TMD-C).

We used an in-house-developed PCF designed for SSFS and fabricated using the stack-and-draw method. The scanning electron microscope (SEM) image of the fiber’s cross section is shown in the inset of Fig. 1(a). The fiber has an elliptical germanium-doped core (doping level in the core equals 18 mol %) with a 4.5 µm major axis and a 2.9 µm minor axis. The core is surrounded by rings of air holes arranged in a honeycomb lattice. The lattice pitch of the air hole structure is 3.0 µm on average, and the diameter of the holes nearest to the core is 0.9 µm on average.

 

Fig. 1. Parameters of the fiber. (a) Calculated mode effective area, (b) measured attenuation, and (c),(d) relative inverse group velocity of the two polarization modes with indications of the characteristic frequency shifts responsible for the given mechanism. Inset in (a) shows the SEM image of fiber’s cross section.

Download Full Size | PDF

We used the cut-back technique to measure fiber attenuation, see Fig. 1(b). In the long wavelength range, attenuation exceeds 1 dB/m for wavelengths longer than 2.12 µm. Using the white-light interferometric method, we measured relative inverse group velocities (RIGV, $\Delta \beta _1^{(x/y)}$) of the two polarization modes in a short fiber section, see Figs. 1(c) and 1(d). The results allowed us to determine the value of group birefringence as approximately −2×10^-4 in the spectral region 1.6–2.15 µm. From the RIGV slope we established chromatic dispersion. For pulses polarized along the slow fiber axis it is 22 ps km⁻¹ nm⁻¹ at the pump wavelength.

The fiber’s geometrical parameters obtained from the SEM image were used in the numerical modeling of the fiber transmission properties developed with the Comsol Multiphysics Wave Optics Module. Simulations performed using the finite element method allowed us to calculate the wavelength dependence of the effective refractive index and mode effective area $A_\mathrm {eff}$, see Fig. 1(a).

To verify the impact of the orthogonal Raman scattering on the pulse propagation and to get an insight into the evolution of the pulse along the propagation distance we performed numerical simulations with a self-developed solver [19] based on the software implemented by Travers et al. [20]. The simulations were based on the set of two-mode coupled nonlinear Schrödinger equations (CNLSE) described in detail in Supplement 1. The nonlinear term includes the co-polarized and orthogonal Raman scattering [8,12,21–23]. In the first place, we were able to turn on–off the polarization mixing term in the numerical experiment to reveal its contribution to the observed processes. Moreover, the spectral dependency of the effective mode area was accounted for by introducing the spectral pseudo-envelope of amplitude as proposed by Laegsgaard [24]. As can be seen in Fig. 1(a), the effective area is almost identical regardless of the polarization, therefore we assumed the same $A_\mathrm {eff}$ dependency for both polarizations in the simulations. Finally, we extrapolated loss dependency to account for short-wavelength Rayleigh scattering [25,26] and water absorption [27] for the numerical simulations.

Figure 2 shows the optical spectra recorded with an optical spectrum analyzer (Yokogawa AQ6376) at the output of a 20-m-long fiber section for different average power levels. The average power was measured before the lens used to couple light into the fiber. The spectra were recorded with a bandpass filter (Thorlabs, FB2000-500) to reduce the intensity of the spectral region below 1.75 µm. We observe that the input pulse forms a soliton, which undergoes Raman SSFS. This process is schematically represented with an arrow in Fig. 1(c). The characteristic frequency of maximal scalar Raman gain $\Delta \omega _{\parallel } \approx {13}\,\mathrm{THz}$ [12]. The soliton shift increases with the power but the soliton can be tuned hardly beyond 2.1 µm due to the abrupt increase of attenuation, see Fig. 1(b). When the input pulse is polarized along the slow axis, see Fig. 2(a), the soliton pulse is generated with well-defined central wavelength. However, when the input pulse is polarized along the fast axis, see Fig. 2(b), the main soliton pulse in fast axis is accompanied with orthogonally polarized components at longer wavelengths. Due to the transmission loss of the fiber, we observed a decrease in solitons intensity above 2.0 µm, moreover, in the case of 100 mW average pump power, the trapped pulse is attenuated below noise level. The polarization extinction ratio for all observed solitons ranges from 9.1 dB to 11.5 dB, see Fig. 2. There is no clear correlation between the polarization purity of the solitons generated and the input pulse power. This suggests that polarization extinction is limited by linear coupling due to the imperfections of the fiber.

 

Fig. 2. Optical spectra recorded at the output of 20-m-long fiber section for increasing pump power. The polarization direction of the input pulse was aligned along the (a) slow and (b) fast axes of the fiber. We used polarizer to separate the signal components polarized along the slow (green) or fast (red) axes.

Download Full Size | PDF

In further study we focused on the spectral pulse dynamic with propagation distance, which was investigated via cutback of the original 20-m-long fiber. The special care was taken in order to keep the fiber launch condition unchanged. Figure 3 presents the spectra registered at the output of optical fiber sections of different lengths: 2 m, 8 m, and 20 m; for the fixed average power of 60 mW. When the input pulse is polarized along the slow axis, there is no additional spectral component generated. On the contrary, when the input pulse is polarized along the fast axis, the slow axis component is present. The spectral positions of the two polarization components are noted in Table 1, as well as the efficiency of SSFS $\eta$, defined as ratio of the soliton power to the total output power measured with the thermal power sensor using the bandpass filter.

 

Fig. 3. Optical spectra recorded for pump power level of 60 mW at the output of fiber sections of different lengths. The polarization direction of the input pulse was aligned along the (a) slow and (b) fast axes of the fiber. Line colors denote slow (green) and fast (red) axes signal components.

Download Full Size | PDF

Table 1. Measured Spectral Positions of Soliton Polarization Components and Efficiencies of Power Conversion for Slow and Fast Axes Excitation on Different Lengths of the Fiber^a

View Table

Figure 3 shows that the slow axis component increases along propagation when the fast axis is excited. This confirms that trapped pulse components in the slow axis appear due to orthogonal Raman scattering, which enables the simultaneous conversion of wavelength and polarization. This process is schematically presented by the arrow in Fig. 1(d). The characteristic frequency of maximal orthogonal Raman gain $\Delta \omega _{\perp } \approx {3}\,\mathrm{THz}$ [12]. In both excitation cases, the solitons shift toward longer wavelengths. During propagation, they narrow and the polarization extinction ratio lowers. The spectral narrowing and decrease of SSFS efficiency $\eta$ are direct consequences of the attenuation. As the soliton energy decreases its duration increases and soliton narrows spectrally.

To verify the contribution of orthogonal Raman scattering in the polarization conversion process, we simulated the pulse propagation for the two considered excitation conditions. We have performed the simulations turning on and off the polarization mixing terms (as described in Supplement 1). To emulate the linear coupling due to imperfections of the fiber, we assumed that the exciting linear polarization is misaligned from the slow/fast axis by 0.1°. The obtained results are presented in Fig. 4. The simulations revealed that the trapped pulse components are present in the output spectra due to linear coupling, but spectral intensity of the trapped pulse is 15 dB lower when the polarization mixing term is turned off. Additionally, the time profiles obtained with simulations show that polarization components are trapped as they coincide in time, see Fig. S1 in Supplement 1. In both excitation cases the fast axis component is located at shorter wavelength and the slow axis component is located at longer wavelength. However, the orthogonal Raman scattering enables the transfer of energy only in one direction from shorter to longer wavelength.

 

Fig. 4. Calculated optical spectra of generated soliton for polarization direction of input pulse aligned along the (a) slow, and (b) fast axes of the fiber for 60 mW pump power assuming 68% coupling efficiency. Line colors denote different polarization components: slow (solid green) and fast (red) axes when mixing term is turned on; slow (dashed magenta) and fast (blue) axes when mixing term is turned off.

Download Full Size | PDF

The difference between the measurements and simulations is related to the linear coupling of polarizations which was not accounted for in the simulations. Consequently, the polarization extinction ratio is higher than observed in experiments. To underline the impact of orthogonal Raman scattering on the polarization conversion process, we performed additional simulation assuming equal distribution of pump pulse between polarization modes. For this excitation condition the linear coupling between polarization modes balances. The results presented in Fig. S2 in Supplement 1 show the crucial role of orthogonal Raman scattering in energy transfer between polarization modes. The experimental confirmation of this observation was beyond the scope of this manuscript and will be investigated in future.

Finally, we analyzed the relation between spectral positions of the main and trapped pulses for fast axis excitation. We summarize the experimental and numerical results achieved for different propagation distances and power levels in Fig. 5. Additionally, we plot the group velocity matched wavelengths (obtained from the RIGV of both polarization modes). The markers indicating the central wavelengths of the pulse polarization components follow the group velocity matching line, which indicates that the orthogonally polarized components have matched group velocities. We note that the group velocity matching was also satisfied in the self-mode conversion investigated by Rishøj et al. [7]. In the case of the polarization conversion reported here, the process takes place on a longer distance than in the mode conversion. This is due to the smaller gain of orthogonal Raman scattering in comparison with the gain of co-polarized Raman scattering. In background of Fig. 5, we plot the gray scale map of normalized orthogonal Raman gain calculated using the Raman response function [12]. It shows that in the fiber under consideration, the pulse components polarized along the slow axis are located at the spectral region for which the strongest orthogonal Raman gain is obtained from the soliton polarized along the fast axis. This confirms that the slow-polarized component originates from the soliton formed initially in fast polarization.

 

Fig. 5. Central wavelengths of two polarization components of the soliton pulse observed after different propagation distance when the input pulse is polarized along the fast axis (markers) and group velocity matched wavelengths for two fiber polarization modes (line). The background intensity indicates normalized orthogonal Raman gain from the fast-polarized pulse component (calculated after Ref. [12]).

Download Full Size | PDF

In this work, we report observation of soliton trapping in a birefringent PCF. We confirmed that orthogonal Raman scattering provides the mechanism of energy transfer between polarization components. Due to the negative sign of group birefringence the slow axis pumping assures higher polarization purity of tuned soliton than the fast axis pumping. This result is important for the tunable sources of polarized light using Raman SSFS in the birefringent PCFs [5]. In the next step, we plan to investigate the influence of the group birefringence value on the soliton trapping. Moreover, by controlling the azimuth angle of linearly polarized light it is possible to control division of pulse power between polarization modes. We presume that the polarization conversion efficiency strongly depends on the initial excitation of the fiber.