Gratings with longitudinal variations in coupling coefficients: super-efficiency and unidirectionality in distributed feedback Raman fiber lasers

We propose a technique to design highly-efficient and -unidirectional DFB Raman fiber lasers based on the engineering of the grating’s coupling coefficient including a π-phase shift position in the fiber. For this purpose, first the ideal intra-cavity signal powers for different pump power levels are determined for given fiber lengths. Then, the sum and difference between the counter-propagating wave intensities at each small segment within fiber lengths are calculated resulting in determining the ideal grating’s coupling functions for co- and contra-directional-pumping. The steady-state behavior of the laser using realistic parameters is finally simulated for modified coupling functions considering the Kerr nonlinearity. For a 10 W co-directional-pumped, ∼1 m long single-mode super-efficient DFB, a ∼50% increase in the laser efficiency, more than 44 dB reduction in contra-directional lasing power, ∼15 times drop in the peak power of intra-cavity signal and ∼38% decrease in the unused-pump power are found, compared to those in a standard DFB with the same coupling-length product. Although an enhanced nonlinear refractive index due to thermal gradient reduces the output power of such lasers, it is shown that the super-efficient laser presents a better performance than the standard one, under such conditions.

During the past years, standard PPS-DFB Raman fiber lasers have been mostly demonstrated for constant-coupling uniform fiber Bragg grating (FBG) with a limited efficiency (power ratio of the output Raman signal to input pump) in which a major part of the pump power remains unused [24,25]. Also, it has already been shown that by pumping an FBG e.g., on the left-hand side, the displacement of the π-phase-shift to the right and left of the center of the grating, results in co-and contra-pumping schemes, as major part of the generated laser power exits from the right-hand side (rhs) and left-hand side (lhs), respectively (increasing unidirectionality at the price of decreasing efficiency [31]). Thus, the standard PPS-DFB Raman fiber laser performance in terms of efficiency and unidirectionality is still limited.
To improve the PPS-DFB laser performance, engineering the grating's coupling distribution function and the π-phase-shift position for a given FBG length, is necessary [7,15,20,31]. The grating's ideal coupling as a function of distance results in extreme power conversion from pump to signal, i.e., efficiency as well as unidirectionality. The π-phase-shift position is also the place at which the coupling function changes sign. The longitudinal variation of the coupling coefficient can be translated into a change in refractive index and inscribed in a fiber core [20,29]. However, it is worth noting that there are significant differences between DFB rare-earth-doped-fiber lasers and DFB Raman fiber lasers. First, the gain mechanism is different. Second, the considerable Kerr nonlinearity of the fiber changes the characteristics of such DFB Raman laser including the lasing wavelength, the intra-cavity power (and consequently the nonlinearity) as well as output power. Third, an increase in nonlinear refractive index induced by thermal gradient is present. Therefore, these different features make a Raman laser a more complex system for which the design should be accompanied by solving the generalized coupled-mode equations. For instance, to increase the small Raman laser power in a PPS-DFB written in a fixed-length fiber (e.g., in a high-loss case), step-strength grating technique [31] has already been used nevertheless, it cannot be applied to general cases and a more universal technique is required. Further, the peak power at the Stokes wavelength within the cavity is increased using the previous technique which is not desirable [30]. Surprisingly, we solve here the problems using a new coupling-engineered technique considering Raman fiber laser features to realize super-efficient PPS-DFBs which may need longer FBGs. Reproducible ultra-long fiber Bragg gratings have been recently realized which provides the possibility for engineering these long gratings [34].
In this paper, first to engineer the coupling of an FBG to increase the laser efficiency and unidirectionality, a design technique for DFB lasers is developed in the linear regime, in section 2. Based on a numerical approach, we calculate the ideal intra-cavity Raman signal powers versus different pump powers for given fiber lengths. The result is used to find the proper sum and difference between the counter-propagating wave intensities, at each small segment within the length of PPS-DFB Raman fiber lasers, to calculate the coupling functions along the fiber lengths for co-and contra-directional pumping schemes. This design is achieved without any previous assumption of the FBG's coupling for a unidirectional operation. In section 3, we include the Kerr nonlinearities to better understand the behavior of Raman DFB lasers and propose a realistic design in the nonlinear regime. The modified coupling profiles for different lengths are applied to the generalized coupled-mode equations in the steady state, considering nonlinear effects, to get a desired length and corresponding coupling profile for an increased output power decreasing an intra-cavity peak power, resulting in a super-efficient laser. The robustness of such a laser against second-mode generation is then described. Finally, the output power reductions of super-efficient and standard PPS-DFB Raman fiber lasers due to an increase in nonlinear refractive index induced by thermal gradient in the fiber are compared and discussed.

FBG design for ideal intra-cavity power in linear regime
The schematic of a DFB Raman fiber laser cavity including an FBG with the length L is shown in figure 1. Assuming slowly-varying envelopes, the standard coupled-mode equations for counter-directional coupling of the electric field amplitudes in a uniform FBG can be written as [35] dA where A + and A − are the forward and backward propagating amplitudes at the distance z, respectively, g is the amplitude gain and κ is the coupling coefficient. Considering two new parameters, S(z) and D(z) as the sum and difference between the counter-propagating wave intensities, they read The intensities are equal to the power divided by the effective area. Multiplying equations (1) and (2) with A + (z) and A − (z), respectively, and taking the sum and difference using equations (3) and (4) and solving for D(z) and κ(z) we have where D (0) is the initial value determined by boundary-conditions (see figure 1). Using equation (5), it can be shown simply that a change in D (z) over an FBG (laser cavity) length, is equal to the total generated or absorbed power as Thus, to generate extreme output power, S (z) should be increased as much as possible in different segments along the length. The boundary condition in a laser resonator at z = 0 dictates that A For the laser to operate, the pump is situated at a wavelength, λ p = λ B − δλ s , where λ B is the Bragg wavelength and δλ s is the Stokes shift so that the Stokes signal wavelength is λ s = λ B . The propagation of the pump, P p and Stokes signal power, P S in a fiber results in pump depletion and Stokes signal amplification, described by the coupled-mode equations [36] where (α p , g p ) and (α s , g s ) are the loss and modified Raman gain coefficients for the pump and signal, respectively. Modified Raman gain coefficients for the signal and pump are given by g s = g R /A eff and g p = g s λ s /λ p , respectively, where g R is the Raman gain coefficient and A eff is the effective mode area of the fiber. D p and G s are the pump depletion and signal amplification functions, respectively and the intra-cavity signal to pump change (per length) ratio is G s P S /D p P p where P S = SA eff . It is worth noting that G s and D p are functions of pump and signal power, respectively. Therefore, for a pump power, by sweeping the signal power, an ideal intra-cavity signal power, can be found resulting in high signal to pump change ratio, for a given length of lossy fiber. The calculations are then performed for different pump powers in order to obtain the ideal intra-cavity signal power for two different fiber lengths which have been shown in figure 2(a). The pump wavelength is chosen to be λ p = 1480 nm resulting in a Stokes wavelength at λ s = 1576 nm. Linear fiber loss is α p = α s which is assumed to be 0.05 dB m −1 in the wavelength region around 1.5 μm [30]. For the fiber, the mode field diameter is ∼4 μm and the Raman gain coefficient is g R = 0.7 × 10 −13 m W −1 for linearly-polarized modes throughout the paper. In figure 2(a), it is seen that for a pump power of 10 W, the ideal intra-cavity signal powers are 1180 W and 527 W for the typical lengths of 42.5 cm and 97.5 cm, respectively, reaching zero for the pump power threshold of ∼1.9 W.
To design the DFB laser for extreme efficiency, the pump power distribution and the corresponding intra-cavity signal power, S(z)A eff are required. We thus need these to calculate D(z) and the coupling using equations (5) and (6). For a given length and an input pump power, an ideal signal power corresponding to P p (0) is found using ideal intra-cavity signal power data for different pump powers (e.g., from figure 2(a)), and considered as P S (0). Considering constant pump and signal power over a small segment of length Δz, the pump at z = Δz is P p (Δz) = P p (0) + ΔP p (0) Δz where ΔP p (0) = −gP S (0) P p (0) − α p P p (0). For this new pump value, we again find the corresponding signal power at Δz using ideal intra-cavity signal power data for various pump powers (e.g., from figure 2(a)) and calculate similarly the new pump and signal power at z = 2Δz. We continue the calculation till the end at z = L to achieve the signal and pump power distribution over the whole length. The inset in figure 2(a) depicts pump power distribution versus length for a pump power of 10 W. The residual pump power at the rhs is almost 1.9 W for a loss of 0.05 dB m −1 . To calculate D(z) using equation (5), we have to define the initial value D (0) according to the boundary condition at z = 0 (see figure 1). Here, for a fully unidirectional laser in the rhs (co-directional pumping), we have D (0) = 0 as no power exits the cavity from the lhs. The second boundary condition on the rhs, expressed as S (L) = D (L), should then be satisfied. Alternatively, for a fully unidirectional laser in the lhs (contra-directional pumping), S (0) = −D (0) so that S (L) = D (L) = 0 is satisfied. Figure 2(b) shows the ideal power, S(z)A eff versus length for the pump power of 10 W and two different lengths of 42.5 cm and 97.5 cm for which the intra-cavity power reaches 1180 W and 527 W, respectively. Thus, a shorter length leads to a higher efficiency at the price of larger intra-cavity peak power. The inset in figure 2(b) demonstrates that the S(z)A eff and D(z)A eff power curves intersect at 42.5 cm and 97.5 cm dictated by the boundary conditions for a fully unidirectional laser in the rhs. For the lengths of 42.5 cm and 97.5 cm, the rhs signal powers are 5.87 W and 5.12 W, respectively.
The engineered PPS-DFB Raman fiber laser consists of a fiber Bragg grating with a length of L and coupling of κ(z). By inserting S(z) and D(z) in equation (6), the proper coupling function, κ(z) between 0 and L = 97.5 cm is calculated. The coupling defines the periodic perturbation and for a sinusoidally varying refractive-index modulation is equal to κ(z) = πΔn(z)/λ B where Δn(z) is the amplitude of the refractive index modulation in the grating [2]. The grating is designed with the period of Λ = λ B /2n eff where n eff is the effective refractive index of the propagating mode. It is worth noting as the coupling and consequently average refractive index of the mode changes, the Bragg wavelength is detuned which is considered later in section 3 via the detuning [1].  Our coupling distribution for extreme efficiency also shows an analogy to rapid adiabatic passage [37]. It is a technique already used in the domain of frequency conversion to solve the need of high efficiency [37] and is being used in other fields of optics [38][39][40][41]. Under this technique, adiabatic variation of the coupling coefficient along the propagation length of (lossless/lossy) fiber allows (full/extreme) transfer of energy from a pump to a Stokes signal. With uniform FBGs, this can be done by changing the index of the modulation pattern, which has a direct influence on the coupling.

FBG design using best realistic couplings considering nonlinearities and thermal gradient
In practice, nonlinearities have an important effect in decreasing the intra-cavity and output power of DFB Raman fiber lasers [29][30][31]. Therefore, to evaluate the laser with realistic parameters, the generalized coupled mode equations in the steady state for the slowly varying amplitudes of the pump wave (A p ) and the forward Stokes(A s + ), and backward Stokes (A s − ) waves, in the presence of the Kerr nonlinearity, should be taken into account which can be written as [2] It should be noted that equations (10) and (11) are not simple extended forms of equations (1) and (2). The squares of the absolute values of the slowly varying amplitudes are equal to the powers of the corresponding waves, i.e., A p 2 = P p and |A s ± | 2 = P s ± . The nonlinear coefficients are defined by γ s = 2πn 2 /λ s A eff and γ p = γ s λ s /λ p where n 2 = 3.2 × 10 −20 m 2 W −1 is the nonlinear refractive index. The detuning of the signal from the Bragg frequency ω B is δ = n(ω − ω B )/c, where ω is the carrier frequency of the slowly varying amplitudes A s ± , n is the refractive index of the fiber core and c is the speed of light in vacuum. Here, we consider a continuous-wave linearly-polarized pump of power P p (0) is launched into the fiber at z = 0 and the polarization state of the pump wave is maintained all the way to z = L. As the pump is far from the Bragg resonance and does not interact with the grating, only the Stokes waves are coupled by the grating. Later for the simulation and design, we solve the coupled-mode equations (9) and (11) using the Runge-Kutta method [31,42] considering the other fiber parameters already utilized in section 2. Also, it should be noted that 2 A p 2 is dominant for the regions where the approximation A s To realize an engineered PPS-DFB Raman fiber laser, a very large coupling at z = 0 is impractical. A solution to the problem in (co-& contra-) pumping configuration is to decrease the unidirectionality by permitting a very small output power in the (lhs & rhs), defined by an increase in the signal power over a short length which is the distance between z = 0 and z = z π . It should be noted that we can select a best coupling for a given short length, z π resulting in maximizing output power (within the used resolution) to prove the efficiency supremacy of engineered PPS-DFB over a standard one with the same strength [31]. Therefore, the ideal coupling profiles for a given length (see figure 3, blue dashed curves), obtained using the aforesaid technique (section 2) and modified by feasible best coupling, κ 1 between z = 0 and z = z π are then applied to equations (9) and (11) to get the largest output power. The best modified coupling profiles for co-and contra-pumping 97.5 cm-long super-efficient PPS-DFB lasers are depicted using black dotted curves in figures 3(a) and (b), respectively. The simulation results have also been reflected in the upper insets in figure 3 showing output power versus z π and κ 1 . At the cost of a small increase in the contra-directional signal powers, the best realistic coupling for the PPS location at 2.5 cm are calculated to be 360 m −1 and 110 m −1 for the largest output power (within the used resolution), in co-& contra-pumping configuration, respectively. Those flat-κ parts are the grating section necessary for the increase in the signal power toward the peak which can be readily obtained with the fiber properties and grating writing technology. Moreover, the zoomed-in parts as the lower insets in figure 3 show the jump of coupling from a large positive to a small negative value at z π = 2.5 cm. Our simulations demonstrate that although the output power is increased from 4.32 W for the co-directional pumping to 4.6 W for the contra-directional pumping, it results in increasing the peak intra-cavity power from 345 W to 542 W. From this point forward, we focus on co-directional pumping generating less intra-cavity power for almost the same output power. Figure 4(a) shows the rhs output power and intra-fiber peak power versus fiber length for 10 W-pumped engineered PPS-DFBs with and without nonlinearities, considering the best coupling for each length. It is seen in figure 4(a) that nonlinearities in general result in decreasing the output power and intra-fiber peak power, however, increasing the fiber length initially leads to less intra-fiber peak power and consequently less nonlinearity which enhances the output power. By reaching the best length, the best output power is attained and for longer lengths the nonlinearity is strong enough to again decrease the output power.  with the strength, κL = 12 where the best output power of 2.88 W (with 12.1 dB unidirectionality), minimum intra-cavity peak power of 5540 W and output unused-pump power of 5.73 W are obtained. Therefore, a 50% increase in the laser efficiency (the power ratio of the output signal to input pump), 44 dB reduction in contra-directional lasing power, 15.3 times drop in intra-cavity signal power and 37.9% decrease in unused pump power are found. Furthermore, figure 4(b) shows the transmission for a 97.5 cm-long super-efficient PPS-DFB without and with a 10 W pumping, where the detuning and distortion in the latter are due to the Kerr nonlinearities. The laser provides single-mode operation with a ∼1.2 pm (∼146 MHz) detuning, when pumped which is 10 times less than that of the standard PPS-DFB (∼1.44 GHz) [31]. The linewidths are calculated as 807 KHz and 976 KHz for the super-efficient and standard PPS-DFB [31], respectively. It is worth noting that keeping the fiber length <1 m ensures that the grating's uniformity is maintained along the length and the coupling of the flat-κ part (e.g., ∼360 m −1 ) can be fabricated in practice.
For both above-mentioned PPS-DFBs, the laser has only one longitudinal mode at 10 W, however, using higher input pump power, i.e., 15 W, results in the generation of the second longitudinal mode. For a 15 W pump power, the fundamental and second-order longitudinal modes in the super-efficient PPS-DFB are shown in figure 5(a), where the modes are distributed more widely within the cavity. For comparison, the fundamental and second-order longitudinal modes in the standard PPS-DFB are shown in figure 5(b), where the modes are distributed narrowly within the cavity with less overlapping. The larger overlap in the engineered design results in gain saturation by the fundamental mode, thus, the higher order mode start lasing but with much less power due to the lack of enough gain. For a 15 W pump power, the laser side-mode suppression ratio [8] and mode frequency-distance (see the insets in figures 5) [31] for the super-efficient and standard PPS-DFB are 10.7 dB/83 MHz and 2.6 dB/659 MHz, respectively. Finally, we have already demonstrated that there is a degradation in the output power of PPS-DFB Raman fiber laser due to an increase in nonlinear refractive index [43] induced by thermal gradient along the FBG [30]. The thermal gradient is modeled considering that a temperature variation will lead to an increase in the refractive index of the fiber and thus an increase in phase. By approximating heating as being linear with local intensity we increased the nonlinear refractive index by an amount of n th 2 = dn dT dT dI to represent its contribution to the effective nonlinear refractive index as n eff 2 = n 2 + n th 2 where T is the temperature, and I is the optical intensity of the lasing mode [30]. The effect of an increased nonlinear refractive index on output power for the 10 W-pumped super-efficient and standard PPS-DFB lasers (used in figures 5) is simulated and shown in figure 6. As can be seen, an increase of up to ∼ 18 times in effective nonlinearity ratio, n eff 2 /n 2 can decrease the output power by nearly an order of magnitude, however, the output power for the former is still more than 50% of the latter, over all nonlinearity ratios. The inset in figure 6 also shows that the intra-cavity peak power decreases with the enhancement of nonlinearity and for the super-efficient PPS-DFB is an order of magnitude smaller than the standard PPS-DFB over the range. There is a good way to estimate the magnitude of thermal nonlinear refractive index and consequently the effective nonlinearity ratio to find the peak and output power in figure 6. By solving the heat-transport equation under steady-state conditions, it can be shown [44] that for a continuous-wave laser beam in a core of radius a, the largest thermal-induced nonlinear refractive index can be approximated as n th 2 = dn dT αa 2 /k where α is the absorption coefficient and k is the thermal conductivity. Assuming α ∼ 0.01 m −1 , k ∼ 1.38 W m −1 K −1 and dn dT ∼ 10 −5 K −1 , the thermally-induced nonlinear refractive index is n th 2 = 2.9 × 10 −19 and effective nonlinearity ratio is n eff 2 /n 2 ∼ 10. The intra-cavity peak and output powers for super-efficient PPS-DFB lasers are then 44 W and 0.97 W, respectively.

Conclusion
A new technique was presented to design highly-efficient and -unidirectional DFB Raman fiber lasers based on a π-phase-shifted coupling-engineered fiber Bragg grating. For a given length and pump power in coand contra-directional-pumping, the sum and difference distributions were calculated resulting in finding the grating's ideal coupling function. The steady-state behavior of the laser was simulated for a realistic modified coupling function to evaluate its real output power and intra-fiber peak power considering nonlinearities. For ∼1 m-long co-directional-pumped, super-efficient PPS-DFB, a considerable increase in the laser efficiency, and significant reduction in contra-directional lasing power, peak power of intra-cavity signal and output unused-pump power are found compared to those in a standard PPS-DFB with the same strength. Moreover, an increased overlap between two modes in the super-efficient design promotes robustness against second mode generation and guarantees the stability of the single-mode for pump powers up to ∼15 W. Although an increased nonlinear refractive index due to thermal gradient decreases the output power of PPS-DFB lasers, the super-efficient laser shows a better performance than the standard one.
We have already overcome several issues related to implementing such varying coupling-coefficient profiles discussed in this article, including a pi-phase shift (i.e., apodized PPS-DFB) for short fibers [28,29] and we hope to implement the current designs in long FBGs using our novel fabrication technique to realize meter long devices [34]. The sensitivity of the laser's performance to parameter deviations is also something under further examination for a future publication.