Experimental characterization of Raman overlaps between mode-groups

Mode-division multiplexing has the potential to further increase data transmission capacity through optical fibers. In addition, distributed Raman amplification is a promising candidate for multi-mode signal amplification due to its desirable noise properties and the possibility of mode-equalized gain. In this paper, we present an experimental characterization of the intermodal Raman intensity overlaps of a few-mode fiber using backward-pumped Raman amplification. By varying the input pump power and the degree of higher order mode-excitation for the pump and the signal in a 10 km long two-mode fiber, we are able to characterize all intermodal Raman intensity overlaps. Using these results, we perform a Raman amplification measurement and demonstrate a mode-differential gain of only 0.25 dB per 10 dB overall gain. This is, to the best of our knowledge, the lowest mode differential gain achieved for amplification of mode division multiplexed signals in a single fiber.


Results
The purpose of the present work is to characterize the intermodal Raman overlaps and use them to achieve a minimal MDG in a backward-pumped Raman fiber amplifier. This is done by coupling the pump light into the fiber in the correct combination of the LP 01 -and LP 11 -modes. As will be discussed in the Methods section below, due to strong mode-coupling, the two-fold quasi-degenerate LP 01 -modes and four-fold quasi-degenerate LP 11 -modes are simply considered as two distinct groups of modes. We carry out two measurements: Firstly, the Raman gain of a continuous wave signal in the LP 01 -mode is measured vs. total pump input power for five different modal compositions of the pump, i.e. different combinations of the LP 01 -and the LP 11 -modes. Secondly, both pump and signal are converted to LP 11 . This data is used to calculate the Raman intensity overlaps relative to the LP 01 -LP 01 -overlap, which is all that is needed to find the correct combination of pump modes.
Raman intensity modal overlaps. Assuming both pump, and signal to be CW sources, the signal power P i s , in spatial mode i, and the counter propagating − P j p, and copropagating pump power + P j p, , in spatial mode j, is governed by 17 where α s and α p are loss coefficients for signal and pump wavelengths λ s and λ p , and g R is related to the spontaneous Raman scattering cross section. Note that γ R and α p,s are assumed mode-independent. The intensity overlap integrals are defined as with I i being the intensity of mode i integrated over the entire fiber cross section. Solving (1) and (2) using the undepleted pump approximation, we arrive at an expression for the on/off gain where L eff = (1 − exp[− α p L])/α p is the effective fiber length and L is the physical fiber length. The setup used is a backwards pumped configuration, where the pump has only two different spatial profiles (corresponding to LP 01 and LP 11 ), so Eq. (4) can be reduced to i i i R eff p ,11 p ,01 p for the signal in mode i, where η p is the degree of conversion of the pump from LP 01 to LP 11 (η p = 0 when all the pump power is in LP 01 , and η p = 1 when all the pump power is in LP 11 ) and P p is the total input pump power (in all modes). Using the setup which is described in the methods section below, 65 measurements were carried out with 5 different conversion degrees and 13 different pump power levels varying from 0 to 1200 mW for each conversion degree. From the expected form of the gain, Eq. (5), we fitted a function of the form   (5) and (6). This result agrees well with the value of 0.48 obtained from simulated mode-profiles provided by the fiber supplier. Subsequently, the signal was coupled to the LP 11 -mode with the highest attainable efficiency, (η p > 0.99), and the pump was converted to the LP 11 -mode with an efficiency of η p = 0.925, see the Methods section for details, and the Raman gain of the LP 11 -signal was measured vs. the input pump power. A linear function of the type is calculated, taking into account the pump conversion degree. The slope obtained from the fit to the LP 11 -LP 11 -data was c 3 = 4.74 dB/W. We assume wavelength independence of the overlap integrals (i.e. that the LP 01 -LP 11 and LP 11 -LP 01 overlaps are nearly identical). By comparison of Eqs (8) and (6) to (5) we note that c 1 = kF 01,01 and c 3 = k(F 01,11 + η p (F 11,11 − F 01,11 )) with γ = k L 10 log (e) 10 R eff . Using Eq. (7) for the ratio F 01,11 /F 01,01 , these two expressions can be rearranged to give . This is compared to the simulated values for LP 11a -LP 11a and LP 11a -LP 11b of 0.72 and 0.24, respectively. The measured overlap is, as expected, an intermediate value that depends on the mode-coupling within the LP 11 mode-group. In Table 1 the measured overlaps are summarized, and in Table 2 the simulated overlaps are shown. Notice that the overlaps are normalized so that the LP 01 -LP 01 -overlap equals one.

Mode-equalized Gain Based on Measured
Overlaps. Since the LP 11 -LP 11 and LP 11 -LP 01 intensity overlaps often turn out to be very similar in FMFs, relatively low differential gain can be obtained by simply launching the pump completely into LP 11 . This was experimentally verified by R. Ryf et al. 6 where a differential gain of 0.5 dB per 10 dB of gain was observed. For the fiber used in this work, such a scheme results in a differential gain of 1 dB per 10 dB of gain as obtained from the data shown in Fig. 2b (the differential gain in the figure is slightly lower since the pump is only converted 95% into LP 11 ). Using our knowledge of the intensity overlap integrals, the condition for equal signal gain across the two signal-modes, G 11 = G 01 , can be written as   In Fig. 2a the results of measuring a signal launched first completely in LP 01 and then completely in LP 11 with a pump conversion of η p = 0.83, i.e. slightly below the optimal value, are shown. From the figure it is clear that very little mode-dependent gain remains (compare with Fig. 1b). The mode-differential gain as a function of the mean gain is seen in Fig. 2b, showing a residual MDG of only 0.25 dB per 10 dB of Raman gain as obtained from the fitted lines. This differential mode gain is, to the best of our knowledge, the lowest that has so far been experimentally demonstrated. The reason for the fluctuation in MDG is most likely due to mode coupling between LP 11a and LP 11b . The LPG preferentially couples to the LP 01 mode that we detect in the optical spectrum analyzer (OSA), as explained in the Methods section. This means that any mode coupling between LP 11a and LP 11b shows up as a small variation in the measured amplified signal. In the η p = 0.83 (blue dot) measurement the back coupling is slightly more unstable compared to the η p = 0.95 (red circle). This is due to the different configuration of the back coupling LPG.

Methods
The intermodal Raman gain is measured using the experimental setup shown in Fig. 3. The setup is a distributed backwards pumped multi-mode Raman amplifier with a CW laser operated at 1550 nm as the signal source, and an unpolarized 1455 nm Raman fiber laser used for optical pumping. The characterized fiber is a 10 km, 2-moded graded-index fiber.
Higher-Order Mode excitation. The excitation of higher-order modes is achieved by use of mechanically induced LPGs, which are created by pressing the fiber between a periodically grooved aluminum block and a rubber pad. This creates a periodic perturbation in the fiber index, which induces mode coupling if the pitch of the induced gratings matches the difference in propagation constants of the modes 18 . Using a broadband supercontinuum source at the signal input the mode-converted wavelengths are observed in the OSA 2 as a drop in the power spectrum due to the FMF to single mode fiber splice working as a mode filter. The effective pitch of the LPG is changed by adjusting the angle of the grooves with respect to the fiber, until maximum mode-conversion is achieved at the signal wavelength. The use of a supercontinuum source for calibration is not strictly necessary if the difference in propagation constant for the modes of interest is known, but it facilitates  the excitation process. Based on the knowledge of the propagation constants the pitch for the pump wavelength was calculated to be 527 μ m, which is in excellent agreement with the 523 μ m pitch experimentally observed at maximum conversion. The LPGs are polarization dependent 18 , so a polarization controller (PC) is used to optimize conversion of the polarized signal source. After propagation through the fiber the signal is converted back to the fundamental mode using a second LPG.
From standard mode-coupling theory the coupling strength between the modes in a step-index fiber is given by 19 where ψ 1,2 are the scalar mode profiles of the fiber. Since the grooves of the mechanical block are only applied to the fiber from one direction, the perturbation Δ ε(r, φ, z) is asymmetric with respect to this direction. Since the LP 01 mode is a circularly symmetric mode, we expect that mainly the LP 11 mode which is spatially asymmetric with respect to the pertubation direction is excited in the induced grating. However, since we use an unpolarized pump, both polarizations of this spatial mode are excited resulting in an almost equal excitation of the four full-vectorial modes (TE 01 , TM 01 , HE 21a and HE 21b ) that constitute the pseudo-LP 11 modes. The strong coupling between these modes is expected to quickly smooth out any difference in the excitation 17 . Thus, following a similar approach as Antonelli et al. 20 , we only consider the excitation of the quasi-degenerate groups of modes, LP 01 and LP 11 , consisting of two and four nearly degenerate modes, respectively. In this regard, the measured overlaps are essentially an average over these groups.

Characterization of fiber under test.
For all measurements the signal power launched is 0.4 mW, and the launched pump power is varied from 0 to 1200 mW. For each pump power the on/off gain is measured by OSA 2 .
The ratio of the LP 01 -LP 01 and LP 01 -LP 11 overlaps is found with the signal in LP 01 and the pump in varying mixtures of both LP 11 and LP 01 by adjusting LPG 2 to the desired pump mode conversion. For the LP 11 -LP 11 gain measurement LPG 1 and PC 1 were adjusted to obtain more than 99% signal conversion, and LPG 2 was adjusted to obtain a maximum of η p = 0.92 pump conversion; The lower pump conversion is due to the pump being unpolarized. The LPG 2 conversion bandwidth is large enough such that, by optimizing PC 2 , 12 dB of the signal is converted back to LP 01 . The back conversion is necessary due to the mode-filtering effect of the single-mode to multi-mode fiber splice. The gain of the back converted signal is the LP 11 -LP 11 gain.
Equal modal gain measurement. To equalize the modal gain, we first adjust LPG 2 so that we are pumping in a combination of the LP 11 and LP 01 modes very close to the optimal value 85% conversion as obtained from the previous measurements, see Eq. (11). We then first adjust LPG 1 and PC 1 to maximize signal conversion (η p > 0.99) and measure the gain of this mode. Then LPG 1 is lifted so that the signal is a pure LP 01 -mode and the gain for this mode is measured. The difference in the gain for these two signal-modes then gives the mode-differential gain.

Conclusion
We have experimentally characterized the intermodal Raman overlaps in a few-mode fiber by varying the launched pump power and the conversion efficiencies of the pump and signal using mechanically induced long-period gratings for mode excitation. The overlap integrals (relative to the LP 01 -LP 01 overlap) for all modal combinations were obtained in this way for a specific few-mode fiber. By use of the obtained overlaps, it was further demonstrated how a mode-differential gain of only 0.25 dB per 10 dB overall gain is obtained by pumping in a specific combination of the LP 11 and LP 01 modes. In the specific few-mode fiber under test, the differential gain was shown to be significantly lower when pumping in the determined combination of modes compared to when pumping only in LP 11 .