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Numerical optimization of multi-wavelength and cascaded Raman fiber lasers
Optics Communications 217 (2003) 233–238 www.elsevier.com/locate/optcom
Numerical optimization of multi-wavelength and cascaded Raman fiber lasers Sven Cierullies *, Hagen Renner, Ernst Brinkmeyer Technical University Hamburg-Harburg, Eissendorfer Strasse 40, 21073 Hamburg, Germany Received 31 October 2002; received in revised form 20 December 2002; accepted 16 January 2003
Abstract We present a model for the simulation and optimization of Raman fiber lasers. Applications include cascaded Raman lasers as well as lasers running on several wavelengths within one Stokes band. The model takes into account forward and backward propagating waves as well as Raman induced interactions between all pump and Stokes lines. Example calculations for both the simulation of a laser and the optimization of the mirror reflectivities for maximum overall output power and equalized laser lines are performed. 2003 Elsevier Science B.V. All rights reserved. PACS: 42.55.W; 42.55.Y Keywords: Raman fiber lasers; Numerical modeling; Optimization design
1. Introduction During the last years, remarkable progress has been made in the development of Raman fiber lasers due to the availability of low-cost, high-power ytterbium double-clad fiber lasers as pump sources. Many papers devoted to cascaded Raman lasers have been published [1,2]. Recently, the development of multi-wavelength Raman lasers that yield output power at several wavelengths within one
*
Corresponding author. Tel.: +49-40-42878-3532; fax: +4940-42878-2860. E-mail addresses: [email protected] (S. Cierullies), [email protected] (H. Renner), [email protected] (E. Brinkmeyer).
Stokes band has gained interest [3–6]. Cascaded Raman lasers have been modelled numerically [7,8] and, in a simplified way, analytically [9]. In these investigations each laser line was pumped only by the neighbouring short-wavelength line. In the present work we develop a model that can be used for simulating both cascaded and multi-wavelength Raman fiber lasers as well as combinations of both. It includes the full Raman interactions between all laser lines. Further, we present a method for the optimization of multi-wavelength lasers.
2. Theoretical model In order to include the interactions between several laser lines within one Stokes band, we
0030-4018/03/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01126-X
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S. Cierullies et al. / Optics Communications 217 (2003) 233–238
extend the models formerly used [7,10]. In these models, it was assumed that each laser line interacts only with the next neighbouring short-wavelength and long-wavelength line. If, however, the spectral distance between the laser lines becomes smaller, Raman induced interactions between all laser lines have to be calculated. So we apply a more general model
a ¼ ða1 ; a2 ; . . . ; an ÞT ; P ¼ ðP1 ; P2 ; . . . ; where T Pn Þ and 0 1 0 kk21 g1;2 kkn1 g1;n B C kkn2 g2;n C B g1;2 0 B C: G ¼ B. ð6Þ C .. .. @ .. A . . g1;n g2;n 0
k¼i X dPi ¼ ai Pi gk;i Pi Pkþ þ Pk dz k¼1 n X kj þ gi;j Pi Pj þ Pj : ki j¼i
One can easily see how the gain matrix G has to be set up for the simulation of different experimental configurations. For cascaded Raman fiber lasers as described in [1,2], where each line i serves as the only pump for the ði þ 1Þth line, only the first superdiagonal and subdiagonal elements are nonzero. In the more general model presented here, interactions between all laser lines are to be calculated, and thus all higher and lower super- and subdiagonal elements have to be set nonzero, according to the experimental Raman gain spectrum of the fiber to be modelled. As mentioned before, the diagonal elements are always equal to zero. With this formulation it is possible to perform the following two different types of calculations: • Simulation: Analysis of a given experimental configuration with known parameters. • Design: Determination of necessary mirror reflectivities to achieve certain power ratios of different laser lines. Calculations for both purposes will be presented in the following.
ð1Þ
Pi ¼ Pi ðzÞ is the power of the forward (+) or backward ()) propagating ith laser line, respectively. Further, ai and ki , where ki1 < ki < kiþ1 , denote the attenuation and the wavelength of this line, respectively, gk;i is the Raman gain coefficient for scattering from the kth to the ith line, and z is the forward propagation distance. In this model we do not distinguish between pump and Stokes lines, since each line can experience an amplification and can serve as a pump, determined by the values of gk;i and gi;j which are zero for i ¼ k or i ¼ j. The number of considered laser lines is determined by the parameter n. Spontaneous scattering is not included, as it is weak and thus only important for excluding a ‘‘trivial solution’’, for which all laser lines are equal to zero. The boundary conditions for a Raman laser with reflecting mirrors such as Bragg gratings at the fiber ends can be written as: P1þ ð0Þ ¼ P0
ðinput pump powerÞ;
ð2Þ
Piþ ð0Þ ¼ Pi ð0Þ Li
ði ¼ 2; . . . ; nÞ;
ð3Þ
Pi ðLÞ ¼ Piþ ðLÞ Ri
ði ¼ 1; . . . ; nÞ;
ð4Þ
where Li and Ri represent the mirror reflectivities for the laser line i at the left (z ¼ 0) and right (z ¼ L) laser ends, respectively. For a better insight into the system, (1) can be rewritten in matrix notation dPi ¼ ½ a G ðP þ þ P Þ i Pi ; dz
ð5Þ
3. Analysis of a multi-wavelength laser The task for the analysis of a given configuration can be formulated as the global minimization problem of an objective function 2 n þ X Pi ðLÞ þ f P2 ð0Þ; . . . ; Pn ð0Þ ¼ ð7Þ Ri ; Piþ ðLÞ i¼2 over the parameter space P2þ ð0Þ; . . . ; Pnþ ð0Þ, where n is the number of laser lines to be modelled and Piþ ð0Þ represents a guessed initial value of Piþ ðzÞ, while the complementary initial value Pi ð0Þ of Pi ðzÞ directly follows from Eq. (3). For this and all following calculations we assumed R1 ¼ 0, i.e., the pump line
S. Cierullies et al. / Optics Communications 217 (2003) 233–238
is not reflected at the right-hand fiber end and the summation starts with i ¼ 2. By varying the values of Piþ ð0Þ the minimum of f is found. It represents the stationary operation of the laser with the backward/forward power ratios at the ends of the fiber close to the reflectivities of the left- and right-hand mirrors. In the present work, the numerical simulations were performed using a differential evolution algorithm [11] implemented in MATLAB [12]. Results for the output powers of a multiwavelength Raman fiber laser as sketched in Fig. 1 are plotted in Fig. 2 in dependence of the input pump power P1þ ð0Þ. The output power of the ith laser line is defined as the difference between the forward and the backward propagating wave at the fiber end, Pout;i ¼ Piþ ðLÞ Pi ðLÞ. The assumed reflectivities are L2 ¼ L3 ¼ L4 ¼ 95% and R2 ¼ 90%, R3 ¼ 70%, R4 ¼ 30%. Although fiber Bragg gratings with reflectivities higher than 95% are commonly used in Raman fiber lasers, the left-hand reflectivities represent an easily achievable value including all splicing losses. (In the calculations, the right-hand reflectivity R1 has been set to zero, i.e., a reflected pump line was not considered. Including a reflected pump line led to higher overall output powers and a higher conversion efficiency, but did not change the observed qualitative behavior.) A fiber attenuation of 2 dB/km for all wavelengths and a maximum gain coefficient of 1 103 ðWmÞ1 have been used. The assumed gain profile is triangular with the maximum at a wavelength shift of 60 nm. One can see that for this configuration it is not possible to obtain equally distributed output powers Pout;i by only varying the input pump power P1þ ð0Þ.
Fig. 1. Example configuration of a multi-wavelength Raman fiber laser operating at three wavelengths in one Stokes band.
235
Fig. 2. Distribution of power among three lines at k2 ¼ 1090 nm, k3 ¼ 1095 nm, and k4 ¼ 1110 nm for different input pump powers P1þ ð0Þ.
4. Design of a multi-wavelength laser The second task, the design of a laser that runs simultaneously on several wavelengths with equalized output power distribution can be achieved by defining another objective function to be minimized. To this aim, the left-hand reflectivities Li are again set to 95%. Also, one of the right-hand gratings has to be assigned a certain reflectivity to obtain a unique solution, e.g., we set R2 ¼ 80%. This value can be changed to obtain different total output powers, as it has already been shown experimentally [5]. In this case, we formulate the task as the minimization problem of the function 2 P ðLÞ f P2 ð0Þ; . . . ; Pn ð0Þ ¼ 2þ R2 P2 ðLÞ n X n X 2 1 þ 2 Pout;i Pout;j ! min !; ð8Þ Pout i¼2 j¼i with the boundary conditions (2) and (3). This objective function requires the ratio of powers of forward and backward propagating waves of line 2 to approach the corresponding right-hand mirror reflectivity R2 , while each line has to yield the same output power Pout;iP ¼ Piþ ðLÞ Pi ðLÞ at z ¼ L. The n parameter Pout ¼ i¼2 Pout;i , describes the total output power of all laser lines. Performing
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Fig. 3. Power distribution along the fiber in a multi-wavelength Raman fiber laser according to Fig. 1 that leads to equalized output powers Pout;2 ¼ Pout;3 ¼ Pout;4 for three lines in one Stokes band.
this calculation with L2 ¼ L3 ¼ L4 ¼ 95% and R2 ¼ 80%, we obtained the optimum reflectivities from the calculated right-hand power ratios ½Pi ðLÞ=Piþ ðLÞ i¼3;4 to be R3 ¼ 65% and R4 ¼ 36%. The resulting power distribution along the fiber with equalized output powers Pout;2 ¼ Pout;3 ¼ Pout;4 is shown in Fig. 3. To show the applicability of the presented design method to recent experimental setups, a multiwavelength laser similar to that presented in [3,4] has been simulated. This laser setup comprises a pump laser with an output power of up to 3.5 W operating at 1100 nm which pumps a nested cavity of four auxiliary cavities to convert the pump power to a wavelength of 1347 nm and a threewavelength laser with equalized output powers at 1427, 1455, and 1480 nm. While the reflectivity of one of the right-hand output couplers with a center wavelength of 1427 nm has been set to 25%, the reflectivity of the other two output couplers had to be determined by the design algorithm. The reflectivities of the high-reflectivity gratings have been set to 99%, and reflection of remaining pump power at the fiber end has been included for this simulation. As they are not given in [3,4], fiber losses of 0.86 dB/km for all wavelengths, a Raman 1 gain coefficient of 1:23 103 ðWmÞ and a fiber length of 600 m have been assumed. The output power of the three equalized laser lines and the total output power are plotted versus the input
Fig. 4. Characteristic of an equalized three-wavelength Raman fiber laser as demonstrated in [3,4]. The output powers of the three output wavelengths are equalized by adjusting the mirror reflectivities for each input pump power Pþ 1 (0). Within the graphical resolution, the three output-power curves cannot be distinguished.
pump power in Fig. 4. Although some of the parameters had to be guessed, the simulation results agree fairly well with the presented experimental data, with regard to the slope efficiency and the threshold pump power. During the simulation, the reflectivities of the output couplers at 1455 and 1480 nm had been re-calculated for each value of the input pump power, and the simulation converged to equalized output powers for all input pump power levels. 5. Optimization for maximum output power The most interesting subject might be to find the mirror reflectivities which lead to the maximum laser output power for equalized laser lines. On first sight, this seems to be a two-stage problem: the condition of equally distributed output powers, which can be fulfilled by an infinite number of different configurations, has to be maintained in first stage. These configurations can be obtained using the numerical minimization described in the previous section for the setup shown in Fig. 1. Assuming the reflectivity of the righthand mirror at k2 is not fixed but a variable parameter, it is possible to calculate the overall
S. Cierullies et al. / Optics Communications 217 (2003) 233–238
237
can see that the simulation indeed found the maximum of this setup with a reflectivity of R2 ¼ 92%. 6. Conclusions
Fig. 5. Overall output power as a function of the right-hand mirror reflectivity R2 . The output power is distributed equally among the laser lines.
output power as a function of the reflectivity R2 . The result can be seen in Fig. 5, where the input pump power P1þ ð0Þ was set to 2.5 W. One can see that the maximum output power is obtained using a reflectivity of R2 ¼ 92%. The task of finding this configuration automatically is now a maximization process (in second stage) over all possible configurations that have previously been selected by solving Eq. (8). This two-stage problem can, however, be solved in a direct one-stage process by defining an appropriate objective function to be minimized. This function has to perform two tasks: maximization of the output power while maintaining equalized power distribution among the laser lines. To obtain a reasonable solution that fulfills both conditions, we choose a function 2 Pn Pn 2 þ e Pout i¼2 j¼i Pout;i Pout;j f ¼ ! min !; 4 Pout ð9Þ where Pout;i is the output power of the ith laser line while Pout is the total output power and e is a variable parameter to be set. This parameter defines, what magnitude of difference between the laser lines is allowed compared to the total output power. An exemplary simulation with e ¼ 0:01 lead to the reflectivities R2 ¼ 92%, R3 ¼ 64%, and R4 ¼ 22%. Comparing this result with Fig. 5 one
In conclusion, we present a model for a multiwavelength Raman fiber laser which can be used both to simulate conventional cascaded Raman fiber lasers as well as multi-wavelength Raman fiber lasers. An optimization scheme which yields the maximum obtainable output power for an equalized power distribution of the laser lines has been presented. This method reduces the two-stage optimization process inherent to the problem to a simpler one-stage process. Each of the objective functions used here represents only a special choice of the formulation of the minimization problem. Other choices with the same minimization properties could be optimized with regard to the numerical convergence. In addition, many other parameters can be included when designing a multi-wavelength Raman fiber laser. With regard to the conversion efficiency, one could adjust the input pump power or the fiber length. Also, a Raman fiber with a larger ratio of the fiber Raman gain to the fiber attenuation [13] can be chosen for the design of an optimized configuration. When one is trying to convert power from a fixed pump wavelength to several fixed wavelengths employing additional auxiliary cavities, the spectral positions of the auxiliary cavities remain to be optimized. Altering the free parameters, one can easily perform such calculations with the presented design method. Each design can easily be adjusted to fibers with different attenuation and Raman gain coefficients by applying the simple scaling rules discussed in [13]. Acknowledgements This work was supported by Agilent Technologies, Germany. References [1] E.M. Dianov et al., Opt. Lett. 25 (6) (2000) 402. [2] M. Prabhu, N.S. Kim, K.-I. Ueda, Opt. Rev. 7 (4) (2000) 297.
238 [3] [4] [5] [6] [7]
S. Cierullies et al. / Optics Communications 217 (2003) 233–238
M.D. Mermelstein et al., in Proc. OFC, 2001, PD3-1. M.D. Mermelstein et al., in Proc. OFC, 2002, TuJ2. C. Headley, in Proc. OFC, 2002, TuB1. X. Normandin et al., in Proc. OFC, 2002, TuB2. M. Rini, I. Cristiani, V. Degiorgio, IEEE J. Quantum Electron. 36 (10) (2000) 1117. [8] S.D. Jackson, P.H. Muir, J. Opt. Soc. Am. B 18 (9) (2001) 1297. [9] I.A. Bufetov, E.M. Dianov, Quantum Electron. 30 (10) (2000) 873.
[10] J. AuYeung, A. Yariv, IEEE J. Quantum Electron. QE-14 (5) (1978) 347. [11] J. Lampinen, Lappeenranta University of Technology, Department of Information Technology, Laboratory of Information Processing, 2001. Available via Internet: http://www.lut.fi/ jlampine/debiblio.htm. [12] Matlab, The MathWorks Inc. http://www.mathworks. com. [13] H. Renner, S. Cierullies, M. Krause, Scaling rules for Raman fiber lasers, paper MF25, to be presented at OFC 2003.
Numerical optimization of multi-wavelength and cascaded Raman fiber lasers Sven Cierullies *, Hagen Renner, Ernst Brinkmeyer Technical University Hamburg-Harburg, Eissendorfer Strasse 40, 21073 Hamburg, Germany Received 31 October 2002; received in revised form 20 December 2002; accepted 16 January 2003
Abstract We present a model for the simulation and optimization of Raman fiber lasers. Applications include cascaded Raman lasers as well as lasers running on several wavelengths within one Stokes band. The model takes into account forward and backward propagating waves as well as Raman induced interactions between all pump and Stokes lines. Example calculations for both the simulation of a laser and the optimization of the mirror reflectivities for maximum overall output power and equalized laser lines are performed. 2003 Elsevier Science B.V. All rights reserved. PACS: 42.55.W; 42.55.Y Keywords: Raman fiber lasers; Numerical modeling; Optimization design
1. Introduction During the last years, remarkable progress has been made in the development of Raman fiber lasers due to the availability of low-cost, high-power ytterbium double-clad fiber lasers as pump sources. Many papers devoted to cascaded Raman lasers have been published [1,2]. Recently, the development of multi-wavelength Raman lasers that yield output power at several wavelengths within one
*
Corresponding author. Tel.: +49-40-42878-3532; fax: +4940-42878-2860. E-mail addresses: [email protected] (S. Cierullies), [email protected] (H. Renner), [email protected] (E. Brinkmeyer).
Stokes band has gained interest [3–6]. Cascaded Raman lasers have been modelled numerically [7,8] and, in a simplified way, analytically [9]. In these investigations each laser line was pumped only by the neighbouring short-wavelength line. In the present work we develop a model that can be used for simulating both cascaded and multi-wavelength Raman fiber lasers as well as combinations of both. It includes the full Raman interactions between all laser lines. Further, we present a method for the optimization of multi-wavelength lasers.
2. Theoretical model In order to include the interactions between several laser lines within one Stokes band, we
0030-4018/03/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01126-X
234
S. Cierullies et al. / Optics Communications 217 (2003) 233–238
extend the models formerly used [7,10]. In these models, it was assumed that each laser line interacts only with the next neighbouring short-wavelength and long-wavelength line. If, however, the spectral distance between the laser lines becomes smaller, Raman induced interactions between all laser lines have to be calculated. So we apply a more general model
a ¼ ða1 ; a2 ; . . . ; an ÞT ; P ¼ ðP1 ; P2 ; . . . ; where T Pn Þ and 0 1 0 kk21 g1;2 kkn1 g1;n B C kkn2 g2;n C B g1;2 0 B C: G ¼ B. ð6Þ C .. .. @ .. A . . g1;n g2;n 0
k¼i X dPi ¼ ai Pi gk;i Pi Pkþ þ Pk dz k¼1 n X kj þ gi;j Pi Pj þ Pj : ki j¼i
One can easily see how the gain matrix G has to be set up for the simulation of different experimental configurations. For cascaded Raman fiber lasers as described in [1,2], where each line i serves as the only pump for the ði þ 1Þth line, only the first superdiagonal and subdiagonal elements are nonzero. In the more general model presented here, interactions between all laser lines are to be calculated, and thus all higher and lower super- and subdiagonal elements have to be set nonzero, according to the experimental Raman gain spectrum of the fiber to be modelled. As mentioned before, the diagonal elements are always equal to zero. With this formulation it is possible to perform the following two different types of calculations: • Simulation: Analysis of a given experimental configuration with known parameters. • Design: Determination of necessary mirror reflectivities to achieve certain power ratios of different laser lines. Calculations for both purposes will be presented in the following.
ð1Þ
Pi ¼ Pi ðzÞ is the power of the forward (+) or backward ()) propagating ith laser line, respectively. Further, ai and ki , where ki1 < ki < kiþ1 , denote the attenuation and the wavelength of this line, respectively, gk;i is the Raman gain coefficient for scattering from the kth to the ith line, and z is the forward propagation distance. In this model we do not distinguish between pump and Stokes lines, since each line can experience an amplification and can serve as a pump, determined by the values of gk;i and gi;j which are zero for i ¼ k or i ¼ j. The number of considered laser lines is determined by the parameter n. Spontaneous scattering is not included, as it is weak and thus only important for excluding a ‘‘trivial solution’’, for which all laser lines are equal to zero. The boundary conditions for a Raman laser with reflecting mirrors such as Bragg gratings at the fiber ends can be written as: P1þ ð0Þ ¼ P0
ðinput pump powerÞ;
ð2Þ
Piþ ð0Þ ¼ Pi ð0Þ Li
ði ¼ 2; . . . ; nÞ;
ð3Þ
Pi ðLÞ ¼ Piþ ðLÞ Ri
ði ¼ 1; . . . ; nÞ;
ð4Þ
where Li and Ri represent the mirror reflectivities for the laser line i at the left (z ¼ 0) and right (z ¼ L) laser ends, respectively. For a better insight into the system, (1) can be rewritten in matrix notation dPi ¼ ½ a G ðP þ þ P Þ i Pi ; dz
ð5Þ
3. Analysis of a multi-wavelength laser The task for the analysis of a given configuration can be formulated as the global minimization problem of an objective function 2 n þ X Pi ðLÞ þ f P2 ð0Þ; . . . ; Pn ð0Þ ¼ ð7Þ Ri ; Piþ ðLÞ i¼2 over the parameter space P2þ ð0Þ; . . . ; Pnþ ð0Þ, where n is the number of laser lines to be modelled and Piþ ð0Þ represents a guessed initial value of Piþ ðzÞ, while the complementary initial value Pi ð0Þ of Pi ðzÞ directly follows from Eq. (3). For this and all following calculations we assumed R1 ¼ 0, i.e., the pump line
S. Cierullies et al. / Optics Communications 217 (2003) 233–238
is not reflected at the right-hand fiber end and the summation starts with i ¼ 2. By varying the values of Piþ ð0Þ the minimum of f is found. It represents the stationary operation of the laser with the backward/forward power ratios at the ends of the fiber close to the reflectivities of the left- and right-hand mirrors. In the present work, the numerical simulations were performed using a differential evolution algorithm [11] implemented in MATLAB [12]. Results for the output powers of a multiwavelength Raman fiber laser as sketched in Fig. 1 are plotted in Fig. 2 in dependence of the input pump power P1þ ð0Þ. The output power of the ith laser line is defined as the difference between the forward and the backward propagating wave at the fiber end, Pout;i ¼ Piþ ðLÞ Pi ðLÞ. The assumed reflectivities are L2 ¼ L3 ¼ L4 ¼ 95% and R2 ¼ 90%, R3 ¼ 70%, R4 ¼ 30%. Although fiber Bragg gratings with reflectivities higher than 95% are commonly used in Raman fiber lasers, the left-hand reflectivities represent an easily achievable value including all splicing losses. (In the calculations, the right-hand reflectivity R1 has been set to zero, i.e., a reflected pump line was not considered. Including a reflected pump line led to higher overall output powers and a higher conversion efficiency, but did not change the observed qualitative behavior.) A fiber attenuation of 2 dB/km for all wavelengths and a maximum gain coefficient of 1 103 ðWmÞ1 have been used. The assumed gain profile is triangular with the maximum at a wavelength shift of 60 nm. One can see that for this configuration it is not possible to obtain equally distributed output powers Pout;i by only varying the input pump power P1þ ð0Þ.
Fig. 1. Example configuration of a multi-wavelength Raman fiber laser operating at three wavelengths in one Stokes band.
235
Fig. 2. Distribution of power among three lines at k2 ¼ 1090 nm, k3 ¼ 1095 nm, and k4 ¼ 1110 nm for different input pump powers P1þ ð0Þ.
4. Design of a multi-wavelength laser The second task, the design of a laser that runs simultaneously on several wavelengths with equalized output power distribution can be achieved by defining another objective function to be minimized. To this aim, the left-hand reflectivities Li are again set to 95%. Also, one of the right-hand gratings has to be assigned a certain reflectivity to obtain a unique solution, e.g., we set R2 ¼ 80%. This value can be changed to obtain different total output powers, as it has already been shown experimentally [5]. In this case, we formulate the task as the minimization problem of the function 2 P ðLÞ f P2 ð0Þ; . . . ; Pn ð0Þ ¼ 2þ R2 P2 ðLÞ n X n X 2 1 þ 2 Pout;i Pout;j ! min !; ð8Þ Pout i¼2 j¼i with the boundary conditions (2) and (3). This objective function requires the ratio of powers of forward and backward propagating waves of line 2 to approach the corresponding right-hand mirror reflectivity R2 , while each line has to yield the same output power Pout;iP ¼ Piþ ðLÞ Pi ðLÞ at z ¼ L. The n parameter Pout ¼ i¼2 Pout;i , describes the total output power of all laser lines. Performing
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Fig. 3. Power distribution along the fiber in a multi-wavelength Raman fiber laser according to Fig. 1 that leads to equalized output powers Pout;2 ¼ Pout;3 ¼ Pout;4 for three lines in one Stokes band.
this calculation with L2 ¼ L3 ¼ L4 ¼ 95% and R2 ¼ 80%, we obtained the optimum reflectivities from the calculated right-hand power ratios ½Pi ðLÞ=Piþ ðLÞ i¼3;4 to be R3 ¼ 65% and R4 ¼ 36%. The resulting power distribution along the fiber with equalized output powers Pout;2 ¼ Pout;3 ¼ Pout;4 is shown in Fig. 3. To show the applicability of the presented design method to recent experimental setups, a multiwavelength laser similar to that presented in [3,4] has been simulated. This laser setup comprises a pump laser with an output power of up to 3.5 W operating at 1100 nm which pumps a nested cavity of four auxiliary cavities to convert the pump power to a wavelength of 1347 nm and a threewavelength laser with equalized output powers at 1427, 1455, and 1480 nm. While the reflectivity of one of the right-hand output couplers with a center wavelength of 1427 nm has been set to 25%, the reflectivity of the other two output couplers had to be determined by the design algorithm. The reflectivities of the high-reflectivity gratings have been set to 99%, and reflection of remaining pump power at the fiber end has been included for this simulation. As they are not given in [3,4], fiber losses of 0.86 dB/km for all wavelengths, a Raman 1 gain coefficient of 1:23 103 ðWmÞ and a fiber length of 600 m have been assumed. The output power of the three equalized laser lines and the total output power are plotted versus the input
Fig. 4. Characteristic of an equalized three-wavelength Raman fiber laser as demonstrated in [3,4]. The output powers of the three output wavelengths are equalized by adjusting the mirror reflectivities for each input pump power Pþ 1 (0). Within the graphical resolution, the three output-power curves cannot be distinguished.
pump power in Fig. 4. Although some of the parameters had to be guessed, the simulation results agree fairly well with the presented experimental data, with regard to the slope efficiency and the threshold pump power. During the simulation, the reflectivities of the output couplers at 1455 and 1480 nm had been re-calculated for each value of the input pump power, and the simulation converged to equalized output powers for all input pump power levels. 5. Optimization for maximum output power The most interesting subject might be to find the mirror reflectivities which lead to the maximum laser output power for equalized laser lines. On first sight, this seems to be a two-stage problem: the condition of equally distributed output powers, which can be fulfilled by an infinite number of different configurations, has to be maintained in first stage. These configurations can be obtained using the numerical minimization described in the previous section for the setup shown in Fig. 1. Assuming the reflectivity of the righthand mirror at k2 is not fixed but a variable parameter, it is possible to calculate the overall
S. Cierullies et al. / Optics Communications 217 (2003) 233–238
237
can see that the simulation indeed found the maximum of this setup with a reflectivity of R2 ¼ 92%. 6. Conclusions
Fig. 5. Overall output power as a function of the right-hand mirror reflectivity R2 . The output power is distributed equally among the laser lines.
output power as a function of the reflectivity R2 . The result can be seen in Fig. 5, where the input pump power P1þ ð0Þ was set to 2.5 W. One can see that the maximum output power is obtained using a reflectivity of R2 ¼ 92%. The task of finding this configuration automatically is now a maximization process (in second stage) over all possible configurations that have previously been selected by solving Eq. (8). This two-stage problem can, however, be solved in a direct one-stage process by defining an appropriate objective function to be minimized. This function has to perform two tasks: maximization of the output power while maintaining equalized power distribution among the laser lines. To obtain a reasonable solution that fulfills both conditions, we choose a function 2 Pn Pn 2 þ e Pout i¼2 j¼i Pout;i Pout;j f ¼ ! min !; 4 Pout ð9Þ where Pout;i is the output power of the ith laser line while Pout is the total output power and e is a variable parameter to be set. This parameter defines, what magnitude of difference between the laser lines is allowed compared to the total output power. An exemplary simulation with e ¼ 0:01 lead to the reflectivities R2 ¼ 92%, R3 ¼ 64%, and R4 ¼ 22%. Comparing this result with Fig. 5 one
In conclusion, we present a model for a multiwavelength Raman fiber laser which can be used both to simulate conventional cascaded Raman fiber lasers as well as multi-wavelength Raman fiber lasers. An optimization scheme which yields the maximum obtainable output power for an equalized power distribution of the laser lines has been presented. This method reduces the two-stage optimization process inherent to the problem to a simpler one-stage process. Each of the objective functions used here represents only a special choice of the formulation of the minimization problem. Other choices with the same minimization properties could be optimized with regard to the numerical convergence. In addition, many other parameters can be included when designing a multi-wavelength Raman fiber laser. With regard to the conversion efficiency, one could adjust the input pump power or the fiber length. Also, a Raman fiber with a larger ratio of the fiber Raman gain to the fiber attenuation [13] can be chosen for the design of an optimized configuration. When one is trying to convert power from a fixed pump wavelength to several fixed wavelengths employing additional auxiliary cavities, the spectral positions of the auxiliary cavities remain to be optimized. Altering the free parameters, one can easily perform such calculations with the presented design method. Each design can easily be adjusted to fibers with different attenuation and Raman gain coefficients by applying the simple scaling rules discussed in [13]. Acknowledgements This work was supported by Agilent Technologies, Germany. References [1] E.M. Dianov et al., Opt. Lett. 25 (6) (2000) 402. [2] M. Prabhu, N.S. Kim, K.-I. Ueda, Opt. Rev. 7 (4) (2000) 297.
238 [3] [4] [5] [6] [7]
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