1. Introduction

Though optical nonlinearity is useful in a number of applications, it is sometimes detrimental. In later cases it is critical to avoid the onset of nonlinear effects by resorting to relatively short signal-propagation distances, since, as universally known, at a fixed optical peak power the nonlinearity threshold is inversely proportional to nonlinear interaction length. This constitutes a general technological constraint in a wide range of applications, such as long-range signal transmission in optical fiber communications, and generation or propagation of high energy and high power signals in fiber lasers. For example, in broad-band continuous-wave and pulsed fiber lasers, where the fiber length is usually limited by the Stimulated Raman Scattering (SRS) [1,2], but where increasing fiber length is needed for more effective heat removal from a fiber at high pumping [2], this trade-off limits the maximum achievable output power.

In this paper, using analytical and numerical-simulation methods, we show that such trade-off can be overcome and that length-independent SRS threshold can be achieved in optical fibers with the Stokes-wave loss. It is important to note that SRS suppression in optical fibers using Stokes-wave loss has been previously proposed and demonstrated with several different techniques [3–9]. However, to the best of our knowledge, it has not been recognized before that it is possible to achieve length-independent SRS threshold when Stokes-wave experiences significant loss. The importance of this finding is that it conceptually enables an entirely new avenue of effectively suppressing nonlinearity compatible with very long fiber lengths.

To practically implement the length-independent SRS threshold for high peak power SRS-free propagation, one needs a fiber structure with a distributive and large Stokes-wave loss, characterized by a spectral profile with a sharp cut-off at short wavelengths to achieve strong discrimination between signal and Stokes waves. Survey of the previously reported methods of SRS suppression through Stokes-wave loss indicates that these techniques are not well suited for achieving length-independent SRS suppression. Indeed, the implementation with long period grating [4] is not distributive, and, therefore, in principle is not compatible with length-independent SRS suppression. The bending of conventional [5] and W-type [6] fibers does not provide with sufficiently sharp short-wavelength cut-off thus making it difficult to differentiate between signal and Stokes waves. As a result these two techniques lead to relatively mild SRS suppression. The dual-hole-assisted fibers [7], filter fibers [8] and all-solid photonic bandgap fibers [9] have sharp wavelength edges, but their suppression magnitudes are also relatively small (usually below 1dB/m). The common limitation for all these techniques caused by the small Stokes-wave loss is that length-independent SRS suppression is weak and, consequently, should occur only in very long fibers according to our estimate - starting from approximately hundreds of meters), which is perhaps the main reason why this length-independent nature of Stokes-wave suppression has not been recognized before. Furthermore, the other common limitation for reported techniques is that they are difficult to implement in large-core fibers, thus making them of limited use in high power laser technology. We show that specially designed large-core Chirally-Coupled-Core (CCC) fibers, which by itself provides improved SRS threshold with a much larger mode field diameter than single mode fibers, can provide with distributive and >10dB/m Stokes-wave loss with sharp short-wavelength cut-off, which makes it capable of sustaining length-independent SRS-free operation at tens-of-kW optical powers starting from few meters of fiber length. Such fibers could facilitate practical implementation of high-power fiber laser and high-power fiber delivery for material processing applications.

2. Length independent SRS threshold with Stokes-wave suppression

In optical fibers, stimulated Raman scattering occurs at sufficiently high pump-wave intensities when the seed Stokes wave P_s(0), consisting of photons generated through spontaneous Raman scattering process at frequencies lower than the pump, starts building up exponentially while propagating with the pump wave along the fiber. This exponential build-up is determined by the Raman gain coefficient g₀, which is proportional to the pump power P_p and propagation length L:

Since both Stokes and pump wave frequencies are separated only by ~13THz, for standard optical fibers one can assume that α_s = α_p = α. In this case, as described in [1], substituting the expression for P_s(0) into Eq. (2), considering typical standard single-mode fibers values for fiber loss coefficient α and effective modal area A_eff, and assuming frequency-independent Raman gain coefficient g₀, one can find an approximate expression for SRS threshold, which is nearly universally used in fiber lasers and nonlinear fiber optics [1,2,10,11]:

From both Eqs. (3) and (4) one can see that when pump and Stokes losses are equal the SRS threshold is indeed inversely proportional to the propagation length.

However, when the Stokes wave experiences larger loss than the pump Δα_s = α_s – α_p > 0, which one can refer to as Stokes-wave suppression, following the same path as described above but with α_s ≠ α_p, one can arrive at the following expression for the SRS threshold in large mode area fibers (30µm 0.06NA step-index, for example):

This leads to an important result: for sufficiently large Stokes suppression Δα_s and long enough propagation length L, the second term becomes dominant in Eq. (5), so the critical power P_cr becomes propagation-length independent. Indeed, with increasing L the first term in Eq. (5) decreases, and eventually becomes much smaller than the second term, so the SRS threshold becomes solely determined by the magnitude of the Stokes suppression Δα_s:

Even though this result might be a bit unexpected, the basic physics behind it is rather straightforward. Let’s rewrite Eq. (2) in a slightly more convenient form:

Since the gain term $e^{P_{cr}\cdot g_0\cdot L/A_{eff}}$ and the suppression term $e^{-\Delta {\alpha }_s\cdot L}$ in Eq. (7) have the same dependence over the length L, the competition between Stokes wave generation and attenuation becomes indeed length-independent. Thus, once Δα_s exceeds P_cr∙g₀/A_eff, as shown by Eq. (6), SRS would never reach threshold regardless of the propagation length L.

To further illustrate this conclusion, we calculate the SRS threshold based on Eq. (5) for 30µm-diameter-core large-mode-area fibers and Stokes suppression levels of 0, 1, 15, 40 and 100 dB/m. Note that the Stokes suppression expressed in terms of dB/m is related to the loss coefficient value as 4.34Δα_s. The calculation results are plotted with color lines in Fig. 1 . Without Stokes suppression, as is indicated by the straight blue line plotted in dB scale, the SRS threshold is inversely proportional to the propagation length and, therefore, can be reached at any low pump power when sufficiently long propagation length is chosen. However, in cases of Stokes suppression at 1, 15, 40, and 100 dB/m levels, shown by the lines in red, purple and cyan correspondingly, SRS threshold power decreases as a straight line on dB scale at short propagation lengths, representing dominance of the first term in Eq. (5). When propagation length gradually increases, each of these three lines gradually converges to a fixed SRS threshold power, representing dominance of the second term in Eq. (5). This long-range SRS threshold power becomes solely determined by the Stokes suppression magnitude Δα_s described by Eq. (6).

 

Fig. 1 The dB-scale of SRS threshold dependence of fiber length for 30μm 0.06NA step-index fibers is plotted. The blue line overlapped with circles, the red line overlapped with squares, the purple line overlapped with triangles, and the cyan line overlapped with diamonds correspond to 0, 1, 15, 40, and 100 dB/m of Stokes suppression respectively. The lines and the symbol points are representing the results from analytical equation and numerical simulation respectively. On the left-down corner, the Raman Stokes gain spectrum of fused silica is shown with a blue curve, while the flattop suppression levels of 1, 15, 40, and 100 dB/m are also shown in the same graph.

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All the above analysis of SRS threshold is based on certain simplifying assumptions, the most important one of which is that we discarded frequency dependence of both the SRS gain and Stokes-wave loss (apart, of course, from taking into account effective SRS gain bandwidth, when calculating Stokes-wave seed power P_s(0) for substitution in Eq. (2). In reality, however, Raman gain spectrum in fused silica has a certain spectrally dependent profile shown in the left-down-corner inset of Fig. 1 [10]. Therefore, to better quantify the conclusions on the length-independent SRS threshold, it is necessary to consider a more comprehensive model with the entire Raman Stokes gain spectrum, which due to the complexity can be only done numerically. Thus, we can use the coupled intensity equations for Raman scattering [10]:

3. SRS suppression in specially designed large-core CCC fibers

The practical challenge of implementing the Stokes wave suppression to achieve length-independent SRS threshold is associated with (i) designing a medium (e.g. an optical fiber) with a precise transmission spectrum to match the required Stokes-wave gain spectrum of an optical fiber (see the left-down-corner inset of Fig. 1), and (ii) achieving sufficiently high Stokes-wave loss so that length-independent SRS suppression would start occurring at practically short fiber lengths. Indeed, to the best of our knowledge, none of existing techniques have demonstrated >10dB/m Stokes-wave suppression. In fact, the all-solid photonic crystal fiber only shows ~1dB/m suppression and the dual-hole-assisted fiber only shows ~0.1dB/m suppression. Thus, according to the plot of 1dB/m suppression in Fig. 1, only a mild increase of the SRS threshold can be achieved, and length-independent nature of such SRS suppression becomes manifest only starting from approximately hundred meter long fibers. In addition, none of the existing techniques have demonstrated the capability to provide large-mode-area effective single-mode performance, which is critical for high power fiber laser and also helpful to increase the SRS threshold intrinsically.

We have developed a new type of large-mode-area effectively single-mode fibers based on Chirally-Coupled-Core (CCC) fiber geometry [12,13], where two fiber cores are deposited within one fiber cladding and placed in optical proximity to form a weakly coupling system. One straight core referred as “central core” is placed along the axis of cylindrical cladding, and the other core referred as “side core” is off-axis and helically winding along the straight central core. Due to the helical symmetry of this chirally coupled system, the eigenmodes inside each core are the so called “helical modes” that are carrying spin and orbital optical angular momentums [12,13]. Thus, the modal coupling between the two cores would show more coupling resonances than the two straight parallel cores in standard parallel waveguides [12,13]. Different from the inter-core coupling resonances between two straight parallel cores [14], CCC inter-core modal coupling resonances are angular-momentum-assisted phase-matching resonances, which are named as quasi-phase-matching (QPM) resonances.

Since every central core mode has different QPM resonances in terms of wavelength position and coupling strength, by manipulating each central mode’s resonance position, resonance strength, and corresponding side modal loss through fiber design parameters, such a condition can be achieved that all higher-order modes are effectively stripped out of the central core and only the fundamental mode is allowed to propagate in such a large mode area fiber [12,13]. However, the central core fundamental mode would also have multiple lossy resonances across the spectrum, which would present themselves as wavelength-selective loss dips in the transmission spectrum. These wavelength-selective loss dips are what can be used to introduce the Stokes-wave suppression. One example of a measured transmission spectrum, showing these fundamental mode loss dips, of a fabricated passive CCC fiber sample is shown with red solid curves in the up-right-corner inset of Fig. 2 (a different transmission spectrum through a different passive CCC fiber sample can be seen in [13]).

 

Fig. 2 The wavelength range at 1085nm~1245nm of one CCC fiber sample’s transmission spectrum with red solid line and vertical axis on the right is shown to match the Raman Stokes gain of pump wavelength at 1085nm which is plotted as a function of wavelength with blue solid line and vertical axis on the left. The broader transmission spectrum of the same CCC fiber sample is shown in the up-right-corner inset.

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Generally speaking, by properly designing the parameters of CCC fibers, one can control the wavelength position and loss magnitude of the fundamental mode’s loss resonance to precisely match the required Stokes wave loss spectrum for the purpose of achieving length-independent SRS threshold. From another perspective, if given a specific fabricated CCC sample such as the one shown in the up-right-corner inset of Fig. 2, one can choose the right pump wavelength to be suppressed for the Raman Stokes-wave gain. In this particular sample (with core size of 30μm), the pump wavelength at 1085nm can be suppressed. In Fig. 2, we plot the Raman Stokes gain of pump wavelength at 1085nm as a function of wavelength with blue solid line and vertical axis on the left, and we also plot the transmission spectrum at 1085nm~1245nm range of this CCC fiber sample with red solid line and vertical axis on the right. It indeed shows that this CCC fiber sample can be used to suppress the Raman stokes wave at the pump wavelength of 1085nm.

This has been verified numerically by incorporating the arbitrary profile of Raman Stokes gain spectrum g_R(λ) and CCC transmission loss Δα_s(λ) of this particular sample into the numerical model based on the coupled intensity equations in Eq. (8). The simulated SRS threshold values for different propagating fiber length are marked by the cyan square symbol in Fig. 3 . It indeed shows that, for this particular Ge-doped CCC fiber sample, very long-distance (>100m) SRS-free propagation can be achieved at pump powers of ~10kW. To compare this numerical result with the previous analysis, the case of no Stokes suppression and the case of 15 dB/m flattop suppression are simulated with the same numerical model and plotted in Fig. 3 as well. The comparison shows that, when fiber length is short, it still follows the trend of no Stokes suppression, while when the fiber length is longer and longer, it starts to follow the trend of approximately 15 dB/m flattop suppression, which means this particular sample can provide the Stokes suppression that is equivalent to approximately 15 dB/m flattop suppression. Therefore, all previous conclusions seem to still hold for the case with arbitrary profile of gain spectrum and suppression spectrum.

 

Fig. 3 The numerical simulation of SRS threshold with Raman Stokes gain profile and CCC Stokes suppression profile is shown with cyan square symbols. For comparison, the numerical simulation results for the case of no stokes suppression with blue solid line and the case of 15 dB/m flat suppression with red solid line are also shown in the same graph.

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4. Discussion and conclusion

In this paper, we have demonstrated the concept of propagation-length-independent SRS threshold using both analytical and numerical analysis. As one suggested practical implementation of this concept, we demonstrate possibility to design CCC-geometry large-core fibers with spectrally tailored fundamental mode transmission, so that length-independent SRS threshold can be achieved at a certain signal wavelength. The measured transmission spectrum of a fabricated CCC sample appears to match the theoretically required Stokes-wave suppression spectral profile. Such approach could be very useful in high power fiber lasers or especially for high power delivery fibers in industrial processing applications.