Stability analysis of pulsed cascaded Raman lasers in dispersion-managed systems

Transcription

1 8 J. Ot. Soc. Am. B / Vol. 8, No. / November Kalyoncu et al. Stability analysis of ulsed cascaded Raman lasers in disersion-managed systems Salih K. Kalyoncu, Shiming Gao, En-Kuang Tien, Yuewang Huang, Dogukan Yildirim, Enver Adas, Stefan Wabnitz, and Ozdal Boyraz,3, * Electrical Engineering & Comuter Science Deartment, University of California, rvine, California 9697, USA Deartment of nformation Engineering, University of Brescia, 53 Brescia, taly 3 Electrical Engineering Deartment, founding member, stanbul Sehir University, stanbul, Turkey *Corresonding author: oboyraz@uci.edu Received July 5, ; revised Setember 9, ; acceted Setember 9, ; osted Setember 6, (Doc. D 5); ublished October 3, We investigate synchronous first and second order ulsed Raman lasers that can achieve frequency sacing of u to cm for CARS microscoy alications. n articular, we focus on analytical and numerical analysis of ulsed stability derived for Raman lasers by using disersion-managed telecom fibers and uming at near 53 nm telecom wavelengths. We show the evolution of the first and second order Stokes signals at the outut for different eak um ower and the net anomalous disersion combinations. We determine the stability condition for disersion-managed synchronous Raman lasers u to second order. The results show that the stable second order Raman Stokes ulses with : W to : W eak ower and s to : s ulse width can be achieved in roosed disersion-managed systems. Otical Society of America OCS codes: 4.35, , NTRODUCTON Wavelength tunable synchronous ulse sources are highly desirable for sectroscoy and otical diagnostics due to their diverse advantages such as broadband oeration, low cost, comactness, and adatability with fibers [ 3]. The common method to generate short ulses in the fiber is the use of otical nonlinearities and soliton ulse shaing in the anomalous disersion regime that results in sectral broadening and ulse comression [4]. However, to generate ultrashort ulses, a broadband gain mechanism is also required. Stimulated Raman scattering with a wide gain bandwidth of 6 THz has an ability to generate femtosecond ulses [5,6]. Additionally, Raman gain can be generated at any wavelength in a nonlinear medium, which facilitates generation of ulses at wavelengths that are not attainable by a conventional laser gain medium. Self-soliton frequency shift (SSFS) of otical ulses in a highly nonlinear fiber (HNLF), such as hotonic crystal fiber, due to the self-raman scattering effect enables tunable femtosecond ulse generation over a range of > nm [7]. Fixed-wavelength Raman lasers have been widely studied for the ast two decades. Recently, much focus has been shifted toward multi-wavelength tunable Raman fiber lasers that can generate outut Stokes ulses in a broad wavelength range by the so called cascaded stimulated Raman scattering [8 ]. For sectroscoy and otical diagnostics, hase coherent and wavelength tunable synchronous ulse sources are of great demand. n articular, in chemistry and biological alications, higher-order cascaded ulsed Raman lasers with u to cm frequency sacing is highly desirable for CARS microscoy. Previous exerimental works have been dedicated to address that requirement [ 4]. n this aer, we roose and analyze the stability of a new configuration for first and the second order ulsed Raman lasers with a frequency tuning range u to cm. We carry out a detailed numerical analysis in order to investigate the stability regime of the generation of Stokes ulses u to the second order in a disersion-managed (DM) fiber ring cavity umed by a 53 nm fiber laser. We will show that stable Stokes ulses can be generated with different um ower levels and lengths of the single mode fiber (SMF) that is included in the DM ma that forms the cavity. A variational analysis, more owerful and elaborate than the one used in [5] and in our own recent related study [6], is also emloyed in order to analytically investigate the steady-state dynamics of Stokes ulses inside the cavity and to redict the outut ulse arameters by solving ordinary differential equations with a eriodic boundary conditions that is a comutationally efficient alternative to time consuming full numerical simulations. The analytical results are comared with simulation results to show good agreement with the derived equations. The results illustrate that the stable second order Raman Stokes ulses with : mw to : W eak ower and s to : s ulse width can be achieved in our DM Raman laser system.. SMULATON SETUP AND VARATONAL EQUATONS n order to generate stable ulsed lasers by the Raman rocess, inventing a roer disersion management of the um, first order, and second order ulses is the most imortant issue. Most of the commercial disersion shifted fibers and SMFs have high disersion values and sloes; hence it is not ossible to maintain a reasonably limited walk-off (which reduces the time overlaing of ulses) through a significant nonlinear interaction length. n order to effectively eliminate such walk-off, we roose the use of the secial DM ma inside the ring laser configuration, which is illustrated in Figs. and. The setu of Fig. can be readily exerimentally 74-34//8-8$5./ Otical Society of America

2 Kalyoncu et al. Vol. 8, No. / November / J. Ot. Soc. Am. B 83 imlemented: in our work we erform a detailed numerical stability analysis in order to otimize its arameters. Our laser is synchronously umed by a 53 nm mode locked laser with s ulse width. The laser cavity is designed as roosed in [5], which mainly consists of two cascaded sections, the DM system as the gain mechanism, and the SMF for ulse shaing. The overlaing of Stokes ulses at the beginning is satisfied by inserting a stretcher (delay line) in one of the ring cavities to match the um reetition rate. Disersion management is formed by cascading HNLF ieces, which have a nearly zero disersion and disersion sloe in a eriodic configuration. n articular, we restrict our analysis to commercially available fibers from different vendors. The characteristics of the HNLFs and the SMF used in the simulations are summarized in Table. n order to raise the net disersion to anomalous regime, SMF is cascaded with the DM system to generate stable soliton-like ulses. The Stokes ulse evolution in our ring laser configuration of Fig. is modeled by assuming that a single (um and Stokes) ulse recirculates in the cavity. Therefore, each round tri in the laser involves roagation through a single eriod of the DM ma as in Fig., which illustrates the disersion rofile for each Stokes order. Stokes ulses are generated through the roagation in the ring laser from the noise level, which is taken as the initial condition. When the system reaches a steady state, the Stokes ulses exactly reeat themselves at each eriod of the ma. Thus, the satial eriodic boundary conditions imly that the initial ulse arameters should be the same as the ones at the end of the cavity. The mutual interaction between the um ulse and first and second order Stokes ulses is governed by a set of three nonlinear Schrödinger (NLS) equations including all rocesses of grou velocity disersion (GVD), self-hase modulation (SPM), cross hase modulation (XPM), walk-off between the ulses, Raman gain, and um deletion [7]: u z þ ν g u t i β u t þ iγju j u ¼ α u g R λ ju λ j u iγðju j þ ju j Þu ; ðþ Disersion(s/km nm) L=8m Length(m) Fig.. DM system for each Stokes ulses. u z þ u ν g t i β ¼ α u þ g R u z þ ν g u Disersion Profile of DM System DM for st Stokes DM for nd Stokes u t þ iγju j u u iγðju j þ ju j Þu ; ju j λ λ ju j t i β L SMF u t þ iγju j u ðþ ¼ α u þ g R ju j u iγðju j þ ju j Þu ; ð3þ where ν g;g;g is the grou velocity, β ;; is the GVD arameter (s =km), α is the fiber loss (km ), g R is the Raman gain coefficient (W km ), γ is the nonlinearity coefficient (W km ), and u ;; is the amlitude of the um, first order Stokes, and the second order Stokes signals, resectively. These governing equations include all the ossible interactions among three otical waves. Four-wave mixing, which cannot build u coherently due to the short length of the HNLFs and the high net disersion of the cavity, as well as the self-raman induced frequency shift, which is negligible for ulses with > s ulse width, are ignored in this analysis. However, time-consuming numerical analysis is required to determine the ulse evolution in the laser cavity and in otical fibers by using these equations. Table. Characteristics of the Fibers Used in the Simulations [] HNLF (þ) Sloe HNLF ( ) Sloe SMF Fig.. Designed setu for first and second order Stokes signals. α: Fiber loss (db=km).9.9. γ: Nonlinearity coefficient.5.5 (=W km) A eff ðμm Þ g R : Raman coefficient (=W km) λ : Zero disersion (nm)/sloe (s=nm km) 53=: 53= : 3=:76 Length (m) 8 8

3 84 J. Ot. Soc. Am. B / Vol. 8, No. / November Kalyoncu et al. Analytical modeling may substantially alleviate comutational requirements. Here, we develoed an analytical model for the roagation of um and the Stokes ulses. The um ulse, which is reetitive at each round tri, is used u in the system and cannot reserve its inut Gaussian enveloe i (ju ð;tþj ¼ P ex h ðt t Þ, where P T is the eak ower, T is the ulse width, and t is the temoral osition). The NLS equation for the um ulse [Eq. ()] can be simlified by neglecting disersive effects (the GVD arameter at the um wavelength is zero and the disersion length is much longer than the cavity length) and by searating sectral (for hase accumulation) and temoral terms (for the ulse shae) so as to solve um roagation analytically. Fiber loss and um deletion (as seen in Eq. (4), which rovides the um temoral evolution) are the dominant terms that affect the ulse shae as follows: u z α u g R λ ju λ j u : The um signal is deleted by the first Stokes signal through Raman interaction. Since the walk-off between the um and the first Stokes signal is very small, saturation in the gain line shows u locally by causing hole burning at the center of the ulse. The evolution of the um ulse is aroximated by solving Eq. (4) as ð4þ ju ðz; tþj ¼ P ex ðt t Þ T Z z λ ex α þ g R ju λ j dz ; ð5þ ju ðz; tþj P ex ðt t Þ T ¼ju ð;tþj ju h ðz; tþj : Z z λ α þ g R ju λ j dz The second term ½u h ðz; tþš in Eq. (6) reresents hole burning at the center of the ulse. The hole is modeled by a Gaussian enveloe with the eak ower (P h ), the ulse width (T h ), and the temoral osition (t h ). All loss mechanisms such as fiber loss and gain deletion lead to the growth of the hole (eak ower). Since the walk-off between the um and the first Stokes ulses (:4 s) is much smaller than the ulse width, the um and the hole are assumed to be temorally cocentered (t h t ). n addition, due to relatively small change in the ulse width of the um and the first Stokes ulses in the DM section at steady state, the ulse width of the hole is aroximated as a combination of um and first Stokes ulse widths. The hole burning is aroximated as ð6þ dp h dz αp λ þ g R P λ P and T h ðzþ T þ = T and t h t : ð7þ The roagation of Stokes ulses is modeled by means of the variational analysis method, which assumes the signal maintains its shae even though its ower, ulse width, and chir changes in a continuous manner and remains the same eriodically after each round tri. The evolution of single ulse arameters in the variational method [8] for the unerturbed NLS equations is evaluated by using the Lagrangian density: L ¼ i u u u u þ β u γ z z t þ u4 : The unerturbed NLS equation is derived from a Lagrangian that is defined as the integration of Lagrangian density with resect to time: L ¼ Z Lðu; u z ;u t ; u ;u z;u t Þdt; where the subscrits z and t corresond to differentiation with resect to distance and time, resectively. The evolution of the ulse arameters is derived by emloying modified Euler-Lagrange equations, taking into account signal to signal interactions such as Raman amlification, XPM, and walk-off as a erturbation to the system. n articular, the hysical effects of XPM and temoral walk-off are dominant for determining the chir evolution, the um deletion, and the gain reduction [9]. The Euler-Lagrange equation reads as L x d Z L ¼ m R u u R dt ; ðþ dz x z x x where x corresonds to any one of the six ulse arameters, such as the energy, the ulse width, the temoral osition, the chir, the central frequency, and the hase, m denotes the imaginary art, and R includes all the erturbation terms analyzed in the right side of the couled NLS equations. The Gaussian ansatz is assumed for both first and second order Stokes ulses as the ulse enveloe [8]: rffiffiffi E ffiffiffi u ¼ T 4 ex ðt t Þ T ð þ icþþiωðt t Þþiφ ; ð8þ ð9þ ðþ where E is the energy, T is the ulse width, t is the temoral osition, C is the chir arameter, Ω is the center frequency, and φ is the hase of the ulse. According to the roosed ansatz, the Lagrangian is defined as L ¼ E dc 4 dz þ C dt T dz þ γe ffiffiffiffiffi þ β ð þ C Þ T 4T þ dϕ dz þ t dω dz þ β Ω : ðþ By substituting the ulse enveloe [Eq. (7)] and the Lagrangian [Eq. (8)] with a Gaussian ansatz into the erturbed Euler-Lagrange equation [Eq. (6)], we derived the modified variational equations for the Stokes ulse arameters, ulse energy (E), ulse width (T), chir (C), temoral osition (t ), and center frequency (Ω) as

4 Kalyoncu et al. Vol. 8, No. / November / J. Ot. Soc. Am. B 85 de dz ¼ αe T ;; ð;t ; Þ þ g RE ffiffiffi J T ;; ð;t ;t Þ λ E ffiffiffi λ T ;; ð;t ;t Þ ; ð3þ dt dz ¼ T de E 4 dz E β C T αe T 3 ;; ðt ;t ; Þ þ g R E ffiffiffi T 3 J ;; ðt ;t ;t Þ λ E ffiffiffi λ T ;; ðt ;t ;t Þ ; ð4þ dc dz ¼ T C de E T dz þ E β ð þ C Þ þ γe T ffiffiffiffiffi T 3 þ αe C T 4 ;; ðt ;t ; Þ g RE C T 4 J ;; ðt ;t ;t Þ λ E ffiffiffi λ T ;; ðt ;t ;t Þ þ γe T J ;; ð;t ;t Þþ E T ;; ð;t ;t Þ 4γE T 4 J ;; ðt ;t ;t Þþ E T ;; ðt ;t ;t Þ ; ð5þ dt dz ¼ w de E dz t E β Ω þ αe T ;; ðt ;t ; Þ g RE ffiffiffi J ;; ðt ;t ;t Þ λ E ffiffiffi λ ;; ðt ;t ;t Þ ; ð6þ T dω dz ¼ αe C E T 3 ;; ðt ;t ; Þ g RE C J ;; ðt ;t ;t Þ λ E ffiffiffi λ ;; ðt ;t ;t Þ T 3 T αe Ω T ;; ð;t ; Þ þ g RE Ω T 3 J ;; ðt ;t ;t Þ λ E ffiffiffi λ T ;; ðt ;t ;t Þ 4γE T 3 J ;; ðt ;t ;t Þþ E T ;; ðt ;t ;t Þ ; ð7þ de dz ¼ αe T ;; ð;t ; Þþ g RE ffiffiffi E T T ;; ð;t ;t Þ; ð8þ dt dz ¼ T de E 4 dz E β C T αe T 3 ;; ðt ;t ; Þ þ g R E E ;; ðt ;t ;t Þ ; ð9þ T 3 T T dc dz ¼ T C de E T dz þ E β ð þ C Þ T 3 þ γe T ffiffiffiffiffi þ αe C T 4 ;; ðt ;t ; Þ g RE C T 4 T ;; ðt ;t ;t Þ þ γe T J ;; ðt ;t ;t Þþ E T ;; ðt ;t ;t Þ 4γE T 4 J ;; ðt ;t ;t Þþ E T ;; ðt ;t ;t Þ ; ðþ dt dz ¼ de E dz t E β Ω þ αe T ;; ðt ;t ; Þ g RE ffiffiffi ;; ðt ;t ;t Þ ; ðþ T dω dz ¼ αe C E T 3 ;; ðt ;t ; Þ g RE C E T 3 ffiffiffi T ;; ðt ;t ;t Þ αe Ω T ;; ð;t ; Þþ g RE Ω E T 3 ffiffiffi T ;; ðt ;t ;t Þ 4γE T 3 J ;; ðt ;t ;t Þþ E T ;; ðt ;t ;t Þ ; ðþ where we define the integrals as a;b;c ðt ;t ;t Þ¼ Z E ðt t Þ a f ðt t Þ b f ðt t Þ c dt; T i ð3þ f ðt t i Þ¼ex ðt t iþ ; where i ¼ ; ; ; ð4þ J a;b;c ðt ;t ;t Þ¼P a;b;c ðt ;t ;t Þ P h a;b;c ðt ;t ;t h Þ: ð5þ Here the subscrits i ¼ ; ; corresond to um, first Stokes, and second Stokes signals, resectively. The relative temoral ositions of the ulse centers of the um and the Stokes signals (the second order Stokes signal is assumed to be the time reference) change mainly due to ulse interactions. Since the frequency shift has a negligible effect, we can ignore the variation in center frequency. However, the temoral shift due to the walk-off effect becomes highly dominant. Thus, we can formulate the change of the ulse temoral ositions due to walk off in a reduced form as dt dz w þ w ; dt dz w ; and dt dz ; ð6þ where w ¼ R λ λ DðλÞdλ D þd ðλ λ Þ w ¼ R λ λ DðλÞdλ D þd ðλ λ Þ stand for the walk-off time between um-first Stokes ( :5 s=m) and first second Stokes ulses ( :9 s=m) er unit length and D is the disersion value (s=nm km).

5 86 J. Ot. Soc. Am. B / Vol. 8, No. / November Kalyoncu et al. 3. SMULATON RESULTS AND STABLTY ANALYSS The main objective of our analysis is to determine if the Stokes ulses converge to DM solitons at the steady state. The slit ste Fourier method () was used to solve the couled NLS Eqs. () (3). We included GVD, SPM, XPM, and Raman gain mechanisms in our simulations and neglect four-wave mixing (because of the relatively short length of the highly nonlinear DM module) and self-raman induced frequency shift (because we consider ulses in the s range) terms. Raman amlification is added as a suort interaction. The first and second order Stokes signals are obtained at the outut for different um ower and SMF length (determines the net anomalous disersion) combinations. Figure 3 illustrates the steady-state Stokes ulses inside the cavity (i.e., before the coulers) and their eak ower evolution in the laser system as an examle of transient behavior toward the steady state. The first Stokes ulse builds u in the system due to roer uming. f the first Stokes ulse has enough energy (i.e., it reaches threshold for the second Stokes ulse to grow), it starts to generate the second Stokes ulse. While the second Stokes ulse builds u in the system, it deletes the first Stokes. Since the system is highly dissiative, the Stokes ulses reach their steady-state level in a damed oscillatory manner. The magnitude of the overshoot, oscillations, and the settling time in the transient resonse deend on the um ower level. The temoral deviation of the ulse center of the first Stokes signal results from the temoral walk-off effect. The enveloe of the Stokes ulse is also characterized by comaring it with a Gaussian ulse with the same arameters. The first Stokes ulse is very well reresented by a Gaussian rofile; however, the tails of the second Stokes ulse are slightly comressed in time so that the Gaussian fitting is less aroriate in this region. The Stokes ulses reeat themselves after each ass (round tri) at steady state. Thus, the is emloyed to roagate the steady-state Stokes ulses for a single ass, so that the intracavity ulse dynamics can be analyzed by extracting the evolution of the ulse arameters. Figure 4 illustrates the evolution of the eak ower and the time width, which is defined as the root mean square (RMS) width of the Stokes ulses throughout the cavity at the steady state. The dynamics of the arameters of each Stokes ulse has a similar behavior. Throughout the DM section of the cavity where all the Raman interactions occur, distributed gain and other nonlinear effects such as SPM and XPM are the dominant terms. Thus, the Stokes ulses are amlified and ositively chired through the DM fibers. Because the disersion length (L D ¼ T =jβ j) in the DM-HNLF is much larger than the amlification (L A ¼ðg R jaj Þ ) and nonlinear lengths (L NL ¼ ðγjaj Þ ), and owing to the net zero disersion of the entire DM ma, linear disersive effects become negligible as seen by the dynamics of the ulse width. However, in the second section of the cavity (standard SMF), since the ulses become searate due to the walk-off, there is virtually no ulse-toulse interaction. Here GVD is the dominant effect throughout the SMF. Since the ulses become highly ositively chired in the DM section of the cavity due to the acquired nonlinear hase, the Stokes ulses first comress and then broaden as seen by the evolutions in Fig. 4 of its eak ower and time width. n order to develo an analytical model to rovide a raid understanding and to exlain the ulse dynamics inside the cavity, a detailed variational analysis was emloyed. The intracavity dynamics of ulse arameters are analyzed in terms of the variational model of Eqs. (3) () as follows. First we use the shooting method in order to satisfy the constraint that the same initial and final values for each ulse arameter are obtained uon a single cavity round tri. Next we exlore the intracavity variation of these arameters by solving the same equations for a given steady-state inut condition over the entire round tri. To comare the results of the variational analysis with the numerical modeling we investigated the ulse Power(W) ulse Gaussian fit (a) Peak Power(W).5.5 (b) 5 5 Time (s) Normalized Time(t/t ) R Power(W) ulse Gaussian fit (c) Peak Power(W) (d) 5 5 Time (s) Normalized Time(t/t ) R Fig. 3. (a) First order Stokes ulse with (b) Gaussian fitting and the evolution of its eak ower and (c) the second order Stokes ulse with (d) Gaussian fitting and the evolution of its eak ower at L SMF ¼ 5 m and P um ¼ :7 W. t R is the round-tri time of the cavity.

6 Kalyoncu et al. Vol. 8, No. / November / J. Ot. Soc. Am. B 87 st Stokes Peak Power(W) Proagation distance(m) (a) nd Stokes Peak Power(W) Proagation distance(m) (b) st Stokes Pulse width(s) Proagation distance(m) (c) nd Stokes Pulse width(s) Proagation distance(m) Fig. 4. ntracavity dynamics of the steady-state ulse arameters, eak ower, and ulse width of first order Stokes (a c) and second order Stokes (b d) at L SMF ¼ 5 m and P um ¼ :7 W. The variational and numerical results are comared. (d) dynamics for a secific SMF length and um ower (L SMF ¼ 5 m and P um ¼ :7 W) as shown again in Fig. 4. The variational equations are quite successful at exlaining the general trend of the ulse arameters at the steady state. Since the variational analysis assumes that the ulse shae does not change throughout the roagation, all erturbations such as Raman gain, um deletion, and the nonlinear chir (induced by XPM) affect the entire ulse by changing its total energy and hase. However, due to temoral walk off and the mismatches between the various ulse widths, these st Stokes Peak Power(W) (a) nd Stokes Peak Power(W) st Stokes Pulse width(s) nd Stokes Pulse width(s) (b) Fig. 5. Steady-state eak ower (a) and ulse width (b c) of the Stokes ulses with resect to eak um ower at L SMF ¼ 5 m. Analytical (solid) and numerical results (dotted) are also illustrated as a comarison. (c)

7 88 J. Ot. Soc. Am. B / Vol. 8, No. / November Kalyoncu et al SMF Length(m) (a) SMF Length(m) (b) SMF Length(m) (c) SMF Length(m) Fig. 6. Contour lots for the eak ower (W) and ulse width (s) of the first order Stokes ulse (a b) and the contour lots for the eak ower (W) and ulse width (s) of the second order Stokes ulse (c d) with resect to the SMF length and eak um ower. (d). erturbations force the ulses to have local changes at different ositions. Thus the variational analysis is accomanied by an intrinsic averaging mechanism that reduces the effect of the gain and the chir (arising mainly from XPM) over the central region of each ulse, which exlains the small deviation between variational and numerical results. n addition to the intracavity ulse dynamics, steady-state ulse arameters such as the eak ower and the time width (RMS width) of the first and the second order Stokes inside the cavity (i.e., before the coulers) that are obtained for different eak um ower levels are extracted by the variational analysis and illustrated in Fig. 5, along with the corresonding numerical results. The overall agreement with the numerical results obtained by the confirms the validity of our variational analysis. We also investigate the stable ulse generation regime, which is defined by the boundaries of either the Stokes threshold on the one hand and the nonlinear induced ulse breaku on the other hand, for various SMF lengths (net disersion) ranging from mto m and for eak um owers from :7 Wto:76 W. Clearly the stability of our ulse solutions in this regime is due to the attractive nature of the strongly dissiative laser ma under consideration. The results are reorted in the contour lots of Fig. 6, which show the eak ower and time width (RMS width) of first and second order Stokes ulses inside the cavity (i.e., before the coulers) that converge at steady state. Figure 6 shows that stable first order Stokes ulses with u to W eak ower and s to : s ulse widths are generated. Similarly, second order Stokes ulses with : Wto: W eak ower and s to : s ulse widths can be obtained. At low um owers that are very close to threshold only, the first order Stokes can be generated. When the um ower is increased, the first order Stokes signal starts to build u in the system and converge to a steady state. f the final eak ower of the first Stokes signal reaches the threshold for second Stokes signal, it generates the second Stokes ulse. At high um owers, due to imbalance between strong nonlinearity and weak disersion, the first Stokes signal starts to break u and the laser system becomes unstable. The robustness to nonlinearity increases roortionally to the net disersion (SMF length). 4. CONCLUSON We investigated the stability of DM synchronous Raman lasers u to second order both analytically and numerically. The variational analysis was conducted in order to derive the differential equations that exlain the intracavity dynamics of Stokes ulses in terms of their ulse arameters. The resulting couled differential equations with eriodic boundary conditions are solved to obtain the Stokes ulse arameters at the steady state. The analytically and numerically derived ulse arameters of the Stokes ulses are comared to illustrate the correctness of the analytical redictions. The results show that stable second order Raman Stokes ulses with : W to : W eak ower and s to : s ulse widths can be achieved in a DM Raman laser system. Results can be further imroved by otimizing the arameters of the disersion management. ACKNOWLEDGMENTS This work is suorted by the DARPA Remoted Analog-to- Digital Converter with De-serialization and Reconstruction (RADER) rogram. REFERENCES. A. A. M. Staring, L. H. Siekman, J. J. M. Binsma, E. J. Jansen, T. V. Dongen, P. J. A. Thijs, M. K. Smit, and B. H. Verbeek, A comact nine-channel multiwavelength laser, EEE Photon. Technol. Lett. 8, 39 4 (996).. M. Zirngibl, C. H. Joyner, C. R. Doerr, L. W. Stulz, and H. M. Presby, An 8-channel multifrequency laser, EEE Photon. Technol. Lett. 8, (996). 3. M. D. Mermelstein, C. Horn, S. Radic, and C. Headley, Sixwavelength Raman fibre laser for C- and L-band Raman

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