## Abstract

We report on theoretical and experimental investigations of gain dynamics in Raman fiber lasers in the frequency range of 1 Hz-1 MHz. An analytical solution of the problem is due to the nonlinear nature of the Raman effect not feasible. Thus, we used a numerical simulation to gain general insights. Experimentally and numerically obtained results for a Raman fiber laser emitting at 1180 nm show good qualitative agreement. We also present a potential physical interpretation of the observed dynamical properties. In addition, we report on an experimental proof-of-principle of a passive pump-to-Stokes RIN suppression scheme for the main Stokes order in cascaded Raman fiber lasers utilizing an additional parasitic Stokes order. Again, results from numerical and experimental studies of a cascaded Raman fiber laser at 1180 nm and 1240 nm show good agreement and confirm the passive pump-to-Stokes RIN suppression at 1180 nm. The dependencies between the resonator design and the parameters of the noise suppression are investigated. In addition, it is shown that the scheme can also be applied to cascaded Raman fiber lasers with more then two Stokes shifts. This opens the possibility to design for example low-noise Raman fiber lasers at 1480 nm to pump low-noise Er^{3+} doped fiber amplifiers.

© 2015 Optical Society of America

## 1. Introduction

Raman fiber lasers (RFLs) and cascaded Raman fiber lasers (CRFLs) are widely used tools to realize light sources at wavelengths for which no active rare-earth dopants are available. The underlying principle is the stimulated Raman scattering (SRS), a third order nonlinear interaction between the optical photons of some optical field and optical phonons of the material. In contrast to the spontaneous Raman effect, which is the conversion of a pump photon either to a Stokes (negative frequency shift) or to an Anti-Stokes (positive frequency shift) photon, the stimulated process can only be realized for the Stokes component. In optical fibers based on silica, an amorphous material, the SRS gain spectrum is quite broad and has a maximum at around 13.2 THz with respect to the frequency of the pump photons. RFLs based on silica fibers have been used for example to realize high power laser sources at 1120 nm [1, 2], which were then used as pump sources for single-frequency Raman amplifiers at 1180 nm and subsequent frequency doubling to 589 nm for laser guide star applications in ground-based space observatories [3–5]. Another example are ultra-long RFLs with cavity lengths of several hundreds of kilometers for sensing applications [6]. Another promising feature of CRFLs is the possibility to set up high-brightness, low-quantum-defect pump sources at 1480 nm for Er^{3+} doped high power fiber amplifiers or lasers [7], as the typical output power levels are far above commercially available laser diodes at 1480 nm. In recent years some on-going power scaling process of CRFLs at 1480 nm took place [8–11], which finally resulted in a record output power of 301 W [12].

Our research is related to laser sources for interferometric gravitational wave detectors (GWDs) and currently all ground-based GWDs are based on lasers at 1064 nm. In order to increase their sensitivity, the next generation of GWDs will most probably also use laser sources at 1.5 *μ*m [13]. A good candidate for these laser sources are single-frequency Er^{3+} doped or Er^{3+}:Yb^{3+} co-doped fiber amplifiers because they can deliver in principle high output power levels with an excellent beam quality. At present, Er^{3+}:Yb^{3+} co-doped fiber amplifiers benefit in terms of output power from commercially available high power pump diodes at 976 nm and a corresponding enhanced absorption of the co-doped Yb^{3+} ions. However, once CRFLs at 1480 nm with similar output power levels become commercially available, pure Er^{3+} doped fiber amplifiers might be an alternative. In addition, the detection of gravitational waves requires laser sources with low initial power noise and, thus, additional external power stabilization is indispensable. Here, Er^{3+} doped fiber amplifiers benefit from the fact that the noise-transfer from the seed and the pump source to the amplified signal can be predicted quite easily, e.g. there exists a full analytical solution [14, 15]. In contrast, the noise-transfer in Er^{3+}:Yb^{3+} co-doped fiber amplifiers is due to the Er^{3+}-to-Yb^{3+} energy transfer more complicated and can not be solved analytically [16]. In an Er^{3+} doped fiber amplifier the pump-to-signal noise-transfer is given by a low pass with a cut-off frequency in the range of 1–10 kHz [15]. Thus, any low-frequency power noise of the pump source is transferred lossless to the amplified signal while the high-frequency power noise is suppressed. Hence, in terms of power noise, Er^{3+} doped fiber amplifiers would benefit from CRFL pump sources at 1480 nm with initial low-frequency power noise.

In this paper we report on gain dynamics in RFLs, in particular in a RFL at 1180 nm. We focus on the frequency range of 1 Hz-1 MHz, as we expect to find the interesting dynamical properties in this domain. In addition, the frequency range up to 100 kHz is also relevant for GWDs [13]. Experimentally and numerically obtained transfer functions are compared to each other qualitatively. Additionally, a potential physical interpretation for the dynamical behavior is given and the influence of the cavity design parameters is analyzed. Furthermore, we report on a passive pump-to-Stokes RIN suppression scheme for CRFLs utilizing an additional parasitic Stokes order. Above the threshold of the parasitic order and by appropriate cavity design the output power of the main Stokes order becomes insensitive to fluctuations of the pump power and, thus, a low-frequency pump-to-Stokes RIN suppression is achieved as it was already proposed by others [17, 18]. To the best of our knowledge, this passive pump-to-Stokes RIN suppression is confirmed by us for the first time experimentally in an 1180 nm and 1240 nm CRFL. In addition, numerical results are again used to investigate the dependencies between the resonator design parameters and the parameters of the noise suppression. At the end of this paper numerical data will be provided that propose that the passive pump-to-Stokes RIN suppression can also be applied for CRFLs at 1480 nm.

## 2. Theoretical model

The general setup of a n-th order CRFL is shown in Fig. 1. In most cases, the pump laser is a linear Yb^{3+} doped fiber laser and the following Raman conversion stage consists of a silica based fiber with Fiber Bragg Gratings (FBGs) at each end. Depending on the actual fiber under use and the corresponding achievable Raman gain, the fiber can have a length from a few tens of meters up to hundreds of kilometers. Inside the Raman conversion stage, the pump laser pumps the first Stokes order via SRS, the first Stokes order pumps the second Stokes order and so on until the last Stokes order is reached. Highly reflective FBGs (HR FBGs) ensure that the power is conserved inside the conversion stage for all intermediate Stokes orders. For the last Stokes order a partially reflective FBG (OC FBG) is used to couple out a fraction of the power.

The classical treatment of SRS in optical fibers yields a system of 2n+1, first order, coupled partial differential equations [19]

*P*

_{0}and the power ${P}_{i}^{\pm}$ of all Stokes orders (

*i*= 1…

*n*). Here, the superscripted

*±*indicates either forward (+) or backward (−) propagation.

*c*

_{0}or

*c*is either the velocity of propagation of the pump or the i-th Stokes order light inside the fiber and

_{i}*α*

_{0}and

*α*is either the linear loss of the pump or the linear loss of the i-th Stokes order light. The term represents the spontaneous Raman effect, with h being the Planck constant and

_{i}*ν*being the frequency of the i-th Stokes order.

_{i}*B*

_{eff,i}is the effective bandwidth of the i-th Stokes order and is given by the bandwidth of the corresponding FBGs. The spontaneous Raman effect depends on the phonon occupancy where

*k*is the Boltzmann constant, T the temperature and Δ

_{b}*ν*the Raman frequency shift (13.2 THz for silica based fibers). If the Raman gain

*g*at some wavelength

_{α}*λ*is known, the Raman gain coefficient for the i-th Stokes order (wavelength

_{α}*λ*) is given by

_{i}The effective interaction area

between the previous and the actual Stokes order is related to the mode field radius [19] of each involved Stokes order and, thus, can be calculated if the core radius*r*and the V number of the fiber under use are known.

_{c}The launched pump power at *z* = 0 and the FBGs for the Stokes orders impose additional boundary conditions

*R*

_{left,i}and

*R*

_{right,i}being the reflectivity of the left and right FBG of the i-th Stokes order. Equations (1) and (7) constitute a boundary value problem (BVP) that can not be solved analytically even for steady state conditions (vanishing time derivatives). A lot of work has been contributed in the past to solve BVPs describing steady-state CRFLs numerically for example by self-written routines [20]. Fortunately, quite a lot ready-made BVP solvers for different programming languages are available nowadays. One example is the FORTRAN shooting-method solver BVP SOLVER [21], for which also a Python wrapper exists [22]. This solver was for example already used to compare experimentally and numerically obtained data for the slopes of a 1480 nm CRFL [23].

Integration of the PDE system (1) along the fiber and corresponding purely time-dependent equations would be the starting point for any analytical investigation regarding the dynamical properties of RFLs or CRFLs. However, to the best of our knowledge, such spatial integration is not analytically feasible for the given case. Thus, we will follow a numerical approach that was already used to determine the characteristic pump-to-stokes noise transfer in RFLs up to frequencies equal to multiples of the cavity’s free spectral range [24].

In order to obtain the transfer functions of a CRFL it is useful to consider a small sinusoidal modulation of the input pump power

*δ*. Then, a corresponding ansatz can be made for the pump power inside the Raman conversion stage and for the power of all Stokes orders

As these equations are linear in *p*_{0} and
${p}_{i}^{\pm}$, the Fourier transformations

Equations (12) and (13) now constitute a 2n+1-dimensional complex BVP. If the steady state solutions are known, Eqs. (12) and (13) can be used to calculate the transfer functions of a CRFL by comparing the input and output magnitudes and phases of the complex solutions. Note that if a specific BVP solver is not capable of solving complex BVPs, Eqs. (12) and (13) can be transferred to a 2(2n+1)-dimensional real BVP by using the real and imaginary parts. For the results presented in the following sections we used the Python wrapped version of BVP SOLVER to solve the steady state BVP (see Eqs. (1) and (7)) of different RFL and CRFL setups and to calculate the corresponding transfer functions for these setups with the BVP given by Eqs. (12) and (13).

## 3. Gain dynamics of a RFL at 1180 nm

Figure 2 shows the experimental setup used to measure the transfer functions of a RFL at 1180 nm. The Yb^{3+} pump laser consisted of a 12 m Yb^{3+} doped double-clad fiber from Nufern with a core diameter of 6 *μ*m and was pumped by a 25 W multimode diode at 976 nm. Lasing at 1117 nm was achieved by a HR FBG with a reflectivity of 99 % and by an OC FBG with a reflectivity of 6%, each with a -3 dB bandwidth of around 1 nm. In order to protect the OC FBG from any residual pump light a cladding light stripper was inserted in between the active fiber and the OC FBG. At a pump power of 18.2 W the Yb^{3+} pump laser delivered an output power of around 7.2 W. The Raman conversion stage consisted of around 50 m non-PM Raman fiber from OFS. A HR FBG with a reflectivity of 99 % and an OC FBG with a reflectivity of 85 %, each with a -3 dB bandwidth of 0.7 nm and a center wavelength of 1180.4 nm were used as a resonator for the Stokes-shifted power. At the output end of the RFL an in-house made 1117/1180 WDM was used to separate the residual pump light from the laser output power at 1180 nm. The threshold pump power for the RFL was 2.2 W and at a pump power of 7.2 W the RFL delivered an output power of 1.2 W at 1180 nm.

For the modulation measurements the current of the multimode 976 nm pump diode was modulated by a self-made current buffer, which was connected in parallel to the DC current driver. A low-ratio tap coupler behind the Yb^{3+} pump laser was used as a reference for the achieved power modulation of the Yb^{3+} pump laser. As the relaxation oscillation frequency of the Yb^{3+} pump laser was between 100 kHz and 500 kHz, a sufficiently large modulation depth up to a modulation frequency of around 1 MHz could be achieved. The pump laser reference and a small fraction of the output power at 1180 nm were detected by fast photodiodes, each with a bandwidth of 150 MHz. Sweeping of the pump power modulation frequency and recording of the corresponding transfer functions was performed with a commercial dynamic signal analyzer.

The measured transfer functions between 1 kHz and 1 MHz for different output power levels of the RFL are presented in Fig. 3. The magnitude of each transfer function is flat at low frequencies and falls off with 10 dB per decade at high frequencies. This behavior corresponds to a low pass and is characterized by a corresponding 3 dB cut-off frequency. Here, the cutoff frequency is in the range of 40 kHz-900 kHz, determined by fitting a low pass to the data. The maximum phase shift of a low pass is -90 degree, which is obviously not the case for the measurements. However, the additional phase shift is explained by the long Raman fiber and corresponding runtime effects at frequencies above 100 kHz. Note that the increasing magnitude and phase shift at very high frequencies around 1 MHz, in particular visible in Fig. 3 for low RFL output power levels, are artifacts, which are caused by the dynamic signal analyzer being unable to keep track of the signal for very small magnitudes. In what follows we will propose a physical interpretation of the observed low pass behavior and analyze the relationships between the cut-off frequency and the cavity design parameters with a numerical simulation.

As mentioned, it is not possible to reduce the CRFL PDE system (see Eq. (1)) to a purely time-dependent problem by analytical integration along the cavity. However, some analogies to traditional lasers, whose dynamical properties are only described by a time-dependent interaction of a total laser field and a cavity-integrated atomic or molecular inversion, are possible. In most cases the transfer function of the total laser field in a traditional laser is given by a behavior known as a relaxation oscillation [25]. Mathematically, it corresponds to a double low pass with complex conjugated cut-off frequencies. However, this is only true if the lifetime associated with the inversion, i.e. the upper laser level, is much larger than the cavity lifetime (class B regime). If this is not the case, the transfer function is given instead by a double low pass with two distinct real cut-off frequencies (class A regime). Although both cut-off frequencies depend on the operation point of the laser, i.e. how far the laser is operated above threshold, the lower cut-off frequency is basically related to the larger lifetime (in most cases the lifetime of the inversion) and the upper one is mainly related to the smaller lifetime (in most cases the lifetime of the cavity).

In a RFL, like in any other laser, the cavity-integrated gain must be constant above the laser threshold in order to always compensate the total losses. Thus, as the gain in a RFL is provided directly by the pump power and not by any inversion, it is the cavity-integrated average pump power that is constant above the threshold of the Stokes order. Hence, as we are interested in the dynamical properties of RFLs, we can think of a RFL as a time-dependent interaction between the Stokes-shifted average laser field and the average pump power, which plays the role of the inversion. Although there exist no mathematical formulas for the associated lifetimes, it seems reasonable to argue that the lifetime of the Stokes-shifted average laser field and the lifetime of the average pump power are of the same magnitude as both will scale with the fiber length. However, due to the FBGs the Stokes photons are confined to the cavity longer than the pump photons and, thus, the lifetime of the Stokes-shifted average laser field is even larger than the lifetime of the average pump power. Hence, one can think of the measured low passes as being the low-frequency parts of double low passes, whereas the second cut-off frequencies lie outside the frequency range accessible in our modulation experiments. In the next paragraph we will use a numerical simulation to show that the cut-off frequency we observed depends in fact on the lifetime of the Stokes-shifted laser field. Thus, in analogy to a traditional laser, the second cut-off frequency may be related to the lifetime of the averaged pump power. This lifetime is at least given by or even smaller than the cavity length divided by the propagation speed of the pump photons. Thus, the corresponding cut-off frequencies (≥6 MHz for 50 m of Raman fiber) would indeed lie outside the frequency range accessible with our experimental setup. However, even if one could measure such high modulation frequencies the subsequent evaluation would be complicated by the fact that one would also measure additional resonances given by the free spectral range of the RFL cavity.

As mentioned, a numerical simulation was used to obtain the transfer functions of a RFL at 1180 nm with similar parameters (Raman fiber length etc.) as defined by the experimental setup. Indeed, the numerically obtained transfer functions show the same characteristic low pass behavior as the experimentally obtained transfer functions. Furthermore, we used the simulation to vary the design parameters of the 1180 nm cavity and to analyze the influences on the cutoff frequencies. Figure 4(a) shows the cut-off frequencies in dependency of the RFL output power for a fixed Raman fiber length of 50 m and a varying reflectivity of the OC FBG. As a larger (smaller) reflectivity yields a longer (shorter) photon lifetime in the cavity it also yields a smaller (larger) cut-off frequency. However, the higher the output power the more pronounced are the differences between the simulated cases. Figure 4(b) shows again the cut-off frequencies in dependency of the RFL output power but this time for fixed reflectivity of 80 % and a varying fiber length. Here, a longer (shorter) fiber length yields a longer (shorter) lifetime and, thus, a smaller (larger) cut-off frequency. Again, the differences between the simulated cases depend on the actual output power.

Although the cavity design parameters and the applied pump power levels were quite similar, the experimentally (40 kHz-900 kHz) and numerically obtained cut-off frequencies (20 kHz-95 kHz) differ significantly. In addition, the experimental cut-off frequencies depend linearly on the RFL output power in contrast to the saturation-like behavior of the numerically obtained cut-off frequencies (see Fig. 4). It is known that in RFLs the intra-cavity spectral power is heavily broadened due to strong Four-Wave-Mixing (FWM) between the longitudinal cavity modes [26]. Thus, in an experiment, the effective reflectivities of the FBGs are lower than the design reflectivities. Furthermore, this effect depends on the intra-cavity power and, therefore, on the actual pump power. Hence, the effective reflectivities in our experimental setup were definitely lower then the specified reflectivities and they were also definitely different for different modulation measurements. These facts certainly explain the discrepancies between the numerically and experimentally obtained cut-off frequencies, as the cut-off frequencies are determined by the effective reflectivities of the FBGs, as just shown in the last paragraph.

## 4. Passive pump-to-Stokes RIN suppression in CRFLs with parasitic Stokes order

Figure 5(a) shows the simulated slope of an 1117 nm-pumped CRFL at 1180 nm with an additional “parasitic” Stokes order at 1240 nm. The Raman fiber length was 30 m and the OC FBGs had a reflectivity of 60 % at 1180 nm and 0.5 % at 1240 nm. Both HR FBGs had a reflectivity of 99 %. Below the threshold of the parasitic Stokes order the output power of the main Stokes order increases linearly with the pump power. However, above the threshold of the parasitic Stokes order the output power of the main Stokes order is nearly constant, i.e. the slope efficiency decreases radically. Here, the output power of the main Stokes order at 1180 nm is insensitive to any pump light power fluctuations. Hence, low-frequency power noise coupled in from the pump laser should be suppressed. This passive pump-to-Stokes RIN suppression is confirmed by the magnitude of the transfer functions of the main Stokes order presented in Fig. 5(b). Below the threshold of the parasitic Stokes order the transfer functions show the typical low pass behavior analyzed in the last section. However, above the threshold the low-frequency noise is suppressed by a factor of *α* up to a cut-off frequency f* _{α}*, which is defined by a 3 dB increase of the magnitude (see Fig. 5(b)). As the inclination of any laser slope defines the strength of the pump-to-signal noise transfer, the suppression factor
$\alpha =10{\mathrm{log}}_{10}(\frac{{\eta}_{2}}{{\eta}_{1}})$ is equal to the change of the slope efficiency below (

*η*

_{1}) and above (

*η*

_{2}) the threshold of the parasitic Stokes order. Thus, the maximum possible pump-to-Stokes RIN suppression is basically just a matter of the cavity design. However, it is not clear at a first glance how the cavity design parameters influence the noise suppression cut-off frequency f

*. This question will be discussed at the end of this section. First, our experimental results are presented in the following and they confirm, to the best of our knowledge, for the first time the passive pump-to-Stokes RIN suppression experimentally.*

_{α}Figure 6 shows the experimental setup used to confirm the passive pump-to-Stokes RIN suppression. The Yb^{3+} doped pump laser was nearly in the same configuration as it was before for the experiments with the 1180 nm RFL. However, this time two 25 W multimode pump diodes were used to pump the laser. Thus, a higher maximum pump power of 27.5 W at 1117 nm could be achieved. As mentioned in the last section, intra-cavity broadening due to FWM results in decreased effective FBG reflectivities. As a maximum noise suppression and, thus, a maximum change of the slope efficiency at the threshold of the parasitic Stokes order is desired, the reflectivities used in the simulation may not be the best choice for an experimental proof-of-principle. Hence, as we are able to manufacture WDMs in contrast to FBGs in-house, different WDMs in loop-mirror configuration were used as output couplers for both Stokes orders. The best results were achieved with a loop-mirror that had by design a reflectivity of 90 % at 1180 nm and 2 % at 1240 nm. As highly reflective counterparts for both Stokes orders we used FBGs with a reflectivity of 99 % and a -3dB bandwidth of around 1 nm. As in the simulations, the Raman fiber had a length of 30 m. At the output end of the CRFL a small fraction of the total output power and a dispersive prism were used to separate the main and the parasitic Stokes order from the residual pump. The mirrors behind the prism introduced wavelength-dependent losses and, thus, the measured power ratios were certainly not the same as in the actual laser output. However, this was tolerable as we were just interested in the change of the slope of the main Stokes order. Figure 7(a) shows the measured power fractions of the main and the parasitic Stokes order behind the prism in dependency of the pump power around the threshold of the parasitic Stokes order. At a pump power of around 11.5 W the parasitic Stokes order started to lase and simultaneously the slope efficiency of the main Stokes order dropped by a factor of around 7.5. Figure 7(b) shows the measured transfer functions of the main Stokes order for different pump power levels below and above the threshold of the parasitic Stokes order. Note that the transfer functions are just shown up to a frequency of 100 kHz, although in principle measurements up to 1 MHz would have been feasible. Unfortunately, in particular for frequencies above 100 kHz, the loop-mirror introduced some non-linear feedback at the wavelength of the pump laser. Thus, as the influence of this feedback on the transfer functions between 100 kHz and 1 MHz is not clear, this part of the transfer functions will not be discussed here. Nevertheless, the passive noise suppression can be clearly identified and the measured suppression factor of around 8.7 dB is as expected equal to the change of the slope efficiency at the threshold. This demonstrates, to the best of our knowledge, for the first time experimentally the passive pump-to-Stokes RIN suppression in a CRFL utilizing a parasitic Stokes order.

As mentioned it is worth to analyze the influences of the cavity design parameters on the noise suppression cut-off frequency f* _{α}*. For this, the cavity design parameters in the numerical simulation have been varied. As a longer (shorter) Raman fiber length will lead to a longer (shorter) lifetime of all (pump, main and parasitic Stokes order) photons one would definitely expect a smaller (larger) cut-off frequency f

*. This is confirmed in Fig. 8(a) for three different Raman fiber lengths (25 m, 30 m and 35 m) and a reflectivity of the OC FBGs of 60 % at 1180 nm and 0.5 % at 1240 nm. The cut-off frequencies have been determined by fitting a high pass to the data in the corresponding frequency range. In addition, also the reflectivity of the OC FBG for the parasitic Stokes order has been varied for a fixed Raman fiber length of 30 m and a fixed reflectivity of the OC FBG of 60 % at 1180 nm (see also Fig. 8(a)). Here, a smaller (larger) reflectivity leads to a larger (smaller) cut-off frequency. As can be seen in Fig. 8(b), also a smaller (larger) reflectivity of the OC FBG at 1180 nm leads to a larger (smaller) cutoff frequency. Thus, the longer the lifetimes associated with the main and the parasitic Stokes order, the smaller the noise cut-off frequency f*

_{α}*.*

_{α}Figure 9(a) shows simulated transfer functions of the parasitic Stokes order at 1240 nm in the frequency range of 1 kHz-1 MHz. As for the results presented in Fig. 5, the Raman fiber length was 30 m, the OC FBGs had a reflectivity of 60 % at 1180 nm and 0.5 % at 1240 nm, and both HR FBGs had a reflectivity of 99 %. Just above the threshold of the parasitic Stokes order the transfer functions appear to be double low passes with two distinct cut-off frequencies (class A regime), in particular good to identify in Fig. 9(a) for a pump power of 5.25 W (black data points). Remarkably, for further increasing pump power a relaxation oscillation peak appears (class B regime), which mathematically still corresponds to a double low pass, but with complex conjugated instead of distinct real cut-off frequencies. In order to analyze this behavior in more detail we performed either fits of a double low pass

with two distinct real cut-off frequencies f_{eff,1} and f_{eff,2} or fits of a double low pass with complex conjugated cut-off frequencies

_{eff,1}, f

_{eff,2}, f

_{real}, and f

_{imag}. As already presumed, just above the threshold the transfer functions are given by double low passes and, moreover, with increased pump power the frequencies f

_{eff,1}and f

_{eff,2}approach each other. At the point where both cut-off frequencies have the same value, the transition from the double low pass (class A) regime to the relaxation oscillation (class B) regime occurs and, thus, for further increased pump power, the transfer functions are then given by double low passes with complex conjugated cut-off frequencies. Unfortunately, such behavior could not be observed in the experiments, mainly due to the maximum available modulation frequency of 100 kHz. However, the origin of this behavior can be understood again by making some analogies to the model of a traditional laser, as it was already used in the last section. In a traditional laser a similar behavior is in principle possible if the lifetime of the photons in the cavity is shorter but not too short in comparison to the lifetime of the inversion. Then, the transition can be driven by just tuning the pump power, e.g. by controlling how far the laser is operated above the threshold. For semiconductor lasers, the reverse transition from the class B to the class A regime has been already observed, but only by changing the cavity length externally [27]. For the case of a CRFL with an additional parasitic Stokes order, the averaged intra-cavity power of the main Stokes order is constant above the threshold of the parasitic Stokes order. Thus, as the 1117 nm pump power in the analysis of the 1180 nm RFL in the last section, it plays the role of the inversion. Due to the reflectivities of the applied FBGs it is reasonable to consider that the lifetime of the averaged intra-cavity power of the parasitic Stokes order (R

_{OC FBG}= 0.5%) is indeed shorter but not too short in comparison to the lifetime of the averaged intra-cavity power of the main Stokes order (R

_{OC FBG}= 60%). Considering this, the observed transition from the double low pass regime to the relaxation oscillation regime can be understood. In addition, as the gain dynamics are our main concern, this is another strong indication for the proposed analogy between the model of a traditional laser and the model of time-dependent interactions between the averaged laser fields in RFLs or CRFLs.

Finally, Fig. 10(a) shows the simulated slope of an 1117 nm-pumped CRFL at 1480 nm with an additional parasitic Stokes order at 1580 nm. The fiber length was 80 m, all HR FBGs had a reflectivity of 99 % and the OC FBGs had a reflectivity of 20 % at 1480 nm and 0.5 % at 1580 nm. Just above the threshold of the parasitic Stokes order the slope the main Stokes order flattens, i.e. the slope efficiency drops dramatically. The corresponding passive pump-to-Stokes RIN suppression at 1480 nm can be observed in the simulated transfer functions in Fig. 10(b). Thus, as it was the case for the CRFL at 1180 nm, the passive pump-to-Stokes RIN suppression should be also applicable experimentally for CRFLs with more than two Stokes shifts. This might be interesting in particular for low-noise CRFLs at 1480 nm as they can be applied as pump sources for low-noise Er^{3+} doped fiber amplifiers or lasers.

## 5. Conclusion

The gain dynamics of Raman fiber lasers in the frequency range of 1 Hz-1 MHz have been analyzed theoretically and experimentally. As a full analytical treatment is not feasible, the time-dependent partial differential equation set describing a Raman fiber laser was Fourier-transformed and the corresponding boundary value problem was solved numerically. Numerically and experimentally obtained transfer functions for an 1180 nm Raman fiber laser pumped at 1117 nm show good qualitative agreement. By making analogies to traditional lasers the simple low-pass behavior of the transfer functions can be understood and the influence of the cavity design parameters on the cut-off frequency has been analyzed with aid of the numerical simulation. By utilizing an additional parasitic Stokes order the slope of a Raman fiber laser can become nearly insensitive to pump power fluctuations above the threshold of the parasitic Stokes order. Thus, in principle a corresponding pump-to-Stokes RIN suppression should be achieved, as it was already proposed by others. This passive pump-to-Stokes RIN suppression was experimentally demonstrated by us for the first time in a Raman fiber laser at 1180 nm with a parasitic Stokes order at 1240 nm. This result will support the design of low-noise Raman fiber lasers. In addition, the impact of the cavity design parameters on the suppression factor and the suppression cut-off frequency have been analyzed within the numerical simulation. Furthermore, numerical data for the 1240 nm Stokes order propose a transition from the class A regime with two distinct real cut-off frequencies to the class B regime with a relaxation oscillation peak. The experimental verification of this effect is yet to be done. In addition, numerical data propose that the passive pump-to-Stokes RIN suppression scheme can also be applied for cascaded Raman fiber lasers with more than one Stokes shift. Thus, 1480 nm cascaded Raman fiber lasers with low initial power noise can be used in the future as pump sources for Er^{3+}-doped amplifiers and lasers with suppressed noise in the low-frequency range. This result is in particular interesting for the next generation of gravitational wave detectors, as they will require laser sources at 1.5 *μ*m with low initial power noise.

## Acknowledgments

This work was supported by the German Research Foundation (DFG) through funding the Cluster of Excellence “Centre for Quantum Engineering and Space-Time Research” (QUEST). In addition, this work was also partially funded by the German Federal Ministry of Education and Research (BMBF) (FKZ: 13N11333).

## References and links

**1. **Y. Feng, L. R. Taylor, and D. B. Calia, “150 W highly-efficient Raman fiber laser,” Opt. Express **17**, 23678–23683 (2011). [CrossRef]

**2. **L. Zhang, H. Jiang, S. Cui, and Y. Feng, “Integrated Ytterbium-Raman fiber amplifier,” Opt. Lett. **39**, 1933–1936 (2014). [CrossRef] [PubMed]

**3. **L. R. Taylor, Y. Feng, and D. B. Calia, “50 W CW visible laser source at 589 nm obtained via frequency doubling of three coherently combined narrow-band Raman fibre amplifiers,” Opt. Express **18**, 8540–8555 (2010). [CrossRef] [PubMed]

**4. **L. Zhang, S. Cui, J. Hu, L. Chen, and Y. Feng, “Power scaling of Raman fiber amplifier based source for laser guide star,” in *Advanced Solid-State Lasers Congress*, P. McManamon, P. Watson, and O. Steinvall, eds., OSA Technical Digest (online) (Optical Society of America, 2013), paper JTh2A.38.

**5. **L. Zhang, H. Jiang, S. Cui, J. Hu, L. Chen, and Y. Feng, “Over 50 W 589 nm single frequency laser by frequency doubling of single Raman fiber amplifier,” in *CLEO: 2014*, OSA Technical Digest (online) (Optical Society of America, 2014), paper SW3N.7.

**6. **H. Martins, M. B. Marques, and O. Frazao, “300 km-ultralong Raman fiber lasers using a distributed mirror for sensing applications,” Opt. Express **19**, 18149–18154 (2011). [CrossRef] [PubMed]

**7. **J. W. Nicholson, “High-power continuous wave Erbium-doped fiber laser pumped by a 1480-nm Raman fiber laser,” *Proc. SPIE*8237, Fiber Lasers IX: Technology, Systems, and Applications, 82370K.

**8. **S. G. Grubb, T. Erdogan, V. Mizrahi, T. Strasser, W. Y. Cheung, W. A. Reed, P. J. Lemaire, A. E. Miller, S. G. Kosinski, G. Nykolak, and P. C. Becke, “High-power 1.48 μ m cascaded Raman laser in germanosilicate fibers,” in *Optical Amplifiers and Their Applications*, Vol. 18 of 1995 OSA Technical Digest Series (Optical Society of America, 1995), paper SaA4. [CrossRef]

**9. **J. W. Nicholson, M. F. Yan, P. Wisk, J. Fleming, F. DiMarcello, E. Monberg, T. Taunay, C. Headley, and D. J. DiGiovanni, “Raman fiber laser with 81 W output power at 1480 nm,” Opt. Lett. **35**, 3069–3071 (2010). [CrossRef] [PubMed]

**10. **V. R. Supradeepa, J. W. Nicholson, C. E. Headley, Y. Lee, B. Palsdottir, and D. Jakobsen, “Cascaded Raman fiber laser at 1480 nm with output power of 104 W,” *Proc. SPIE*8237, Fiber Lasers IX: Technology, Systems, and Applications, 82370J

**11. **V. R. Supradeepa, J. W. Nichsolson, C. E. Headley, M. F. Yan, B. Palsdottir, and D. Jakobsen, “A high efficiency architecture for cascaded Raman fiber lasers,” Opt. Express **21**, 7148–7155 (2013). [CrossRef] [PubMed]

**12. **V. R. Supradeepa and J. W. Nicholson, “Power scaling of high-efficiency 1.5 m cascaded Raman fiber lasers,” Opt. Lett. **38**, 2538–2541 (2013). [CrossRef] [PubMed]

**13. **M. Punturo, *et al.*, “The third generation of gravitational wave observatories and their science reach,” Class. Quantum Grav. **27**, 084007 (2010). [CrossRef]

**14. **S. Novak and A. Moesle, “Analytic model for gain modulation in EDFAs,” J. Lightwave Technol. **20**, 975–985 (2002). [CrossRef]

**15. **H. Tuennermann, J. Neumann, D. Kracht, and P. Wessels, “Gain dynamics and refractive index changes in fiber amplifiers: a frequency domain approach,” Opt. Express **20**(12), 13539–13550 (2012). [CrossRef]

**16. **M. Steinke, J. Neumann, D. Kracht, and P. Wessels, “Gain dynamics in Er^{3+}:Yb^{3+} co-doped fiber amplifiers,” accepted for publication in Optics Express.

**17. **G. Sun, D. Hwang, and Y. Chung, “Stabilization of the output power from a Raman fiber laser by generated-amplified spontaneous emission,” Laser Phys. **18**(10), 1188–1191 (2008). [CrossRef]

**18. **G. Sun, A. Lin, D. Hwang, W.-T. Han, and Y. Chung, “Gain-clamped discrete Raman amplifier with suppressed low-frequency relative intensity noise pump-to-signal transfer,” Laser Phys. **18**(10), 1192–1195 (2008). [CrossRef]

**19. **G. P. Agrawal, *Nonlinear Fiber Optics*, 4th ed. (Academic, 2007).

**20. **S. D. Jackson and P. H. Muir, “Theory and numerical simulation of nth-order cascaded Raman fiber lasers,” J. Opt. Soc. Am. B **18**, 1297–1306 (2001). [CrossRef]

**21. **L. F. Shampine, P. H. Muir, and H. Xu, “A user-friendly Fortran BVP solver,” J. Numer. Anal. Indust. Appl. Math. **1**, 201–217 (2006).

**22. **J. Salvatier, “scikits.bvp solver 1.1,” http://pythonhosted.org/scikits.bvp_solver.

**23. **M. Steinke, E. Schreiber, D. Kracht, J. Neumann, and P. Wessels, “Development of a cascaded Raman fiber laser with 6.5 W output power at 1480 nm supported by detailed numerical simulations,” in 2013 Conference on Lasers and Electro-Optics - International Quantum Electronics Conference, (Optical Society of America, 2013), paper CJ-P.16.

**24. **M. Krause, S. Cierullies, H. Renner, and E. Brinkmeyer, “Pump-to-Stokes RIN transfer in Raman fiber lasers and its impact on the performance of co-pumped Raman amplifiers,” Opt. Commun. **260**, 656–661 (2006). [CrossRef]

**25. **A. E. Siegman, *Lasers* (University Science Books, 1986).

**26. **S. A. Babin, D. V. Churkin, A. E. Ismagulov, S. I. Kablukov, and E. V. Podivilov, “Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser,” J. Opt. Soc. Am. B **24**, 1729–1738 (2007). [CrossRef]

**27. **G. Baili, M. Alouini, T. Malherbe, D. Dolfi, I. Sagnes, and F. Bretenaker, “Direct observation of the class-B to class-A transition in the dynamical behavior of a semiconductor laser,” Europhys. Lett. **87**, 44005 (2009). [CrossRef]