Theory of energy evolution in laser resonators with saturated gain and non-saturated loss

Transcription

1 Theory of energy evolution in laer reonator with urated gain and non-urated lo S. K. Turityn Photonic Reearch Group, Department of lectronic ngineering, Aton Univerity, Birmingham B4 7T, Abtract: Theory of the energy evolution in laer reonator with urated gain and non-urated lo i reviited. An explicit analytical expreion for the output energy/average power in term of the gain uration energy, cavity lo and mall ignal gain parameter i derived for a ring cavity configuration. 9 Optical Society of America OCIS code: (9.436) Nonlinear optic, device; (4.343) Laer theory Reference and Link. A.. Siegman, Laer (Univerity Science Book, 986).. L. F. Mollenauer, and J. P. Gordon, Soliton in Optical Fiber: Fundamental and Application (Academic Pre (6). 3. A. C. Newell, and J. V. Moloney, Nonlinear Optic (Addion-Weley Publihing Company, Redwood City,CA, 99). 4. A. Haegawa, and Y. Kodama, Soliton in Optical Communication (Clarendon, Oxford, 995). 5. V.. Zakharov, and. S. Wabnitz, Optical Soliton: Theoretical Challenge and Indutrial Perpective (Springer-Verlag, Heidelberg, 998). 6. H. A. Hau, Theory of mode locking with a low urable aborber, I J. Quantum lectron. (9), (975). 7. H. A. Hau, J. G. Fujimoto, and. P. Ippen, Analytic theory of additive pule and Kerr len mode locking, I J. Quantum lectron. 8(), (99), Sh. Namiki,. P. Ippen, H. A. Hau, and C. X. Yu, C. X. Yu, nergy rate equation for mode-locked laer, J. Opt. Soc. Am. B 4(8), 99 (997). 8. A. M. Dunlop, W. J. Firth, D. R. Heatley, and. M. Wright, Generalized mean-field or mater equation for nonlinear cavitie with tranvere effect, Opt. Lett. (), (996). 9. J. N. Kutz, Mode-locked oliton laer, SIAM Rev. 48(4), (6).. N. Akhmediev, and A. Ankiewicz, ed., Diipative Soliton, Lecture Note in Phyic, (Springer, Berlin- Heidelberg, 5) Vol N. Akhmediev, and A. Ankiewicz, ed., Diipative Soliton: From optic to biology and medicine, Lecture Note in Phyic, (Springer, Berlin-Heidelberg, 8). Vol J. M. Soto-Crepo, N. N. Akhmediev, V. V. Afanajev, and S. Wabnitz, Pule olution of the cubic-quintic complex Ginzburg-Landau equation in the cae of normal diperion, Phy. Rev. Stat. Phy. Plama Fluid Relat. Interdicip. Topic 55(4), (997). 3. A. I. Chernykh, and S. K. Turityn, Soliton and collape regime of pule generation in paively mode-locking laer ytem, Opt. Lett. (4), 398 (995). 4. L. Kramer,. A. Kuznetov, S. Popp, and S. K. Turityn, Optical pule collape in defocuing active medium, Pima ZTF 6, (995) (, JTP Lett. 6, 94 (995). 5. I. S. Aranon, and L. Kramer, The world of the complex Ginzburg Landau equation, Rev. Mod. Phy. 74(), (). 6. N. N. Roanov, Spatial Hyterei and Optical Pattern (Springer, Berlin, ). 7. W. W. Rigrod, Saturation effect in high-gain laer, J. Appl. Phy. 36(8), (965). 8. A. Chong, W. H. Renninger, and F. W. Wie, "All-normal-diperion femtoecond fiber laer with pule energy above nj," Opt. Lett. 3, 48-4 (7). Introduction There exit a wide variety of laer ytem that cater to a range of application in very divere area of cience and indutry. However, depite multiplicity of configuration, material bae and application, the underlying operational and deign principle appear to be the ame from one device to another, and, thu, imilar baic model can be applied to rather different familie of laer []. Performance and operation of many modern high power/energy advanced laer are determined by a rather complex interplay between a number of phyical effect that include gain, lo, diperion and nonlinearity. ffect of uration, the pectral #6 - $5. USD Received 9 May 9; revied Jun 9; accepted Jun 9; publihed 9 Jun 9 (C) 9 OSA 6 July 9 / Vol. 7, No. 4 / OPTICS XPRSS 898

2 dependence of gain and lo, and the nonlinear dynamic of radiation in the laer cavity make uch laer even more complex phyical ytem. Therefore, deign optimization and modeling of uch laer preent a multi-parametric nonlinear problem and any analytical or emi-analytical reult are very important to reduce the number of optimization parameter and calibrate the model coefficient for comparion with experimental data. Spatial evolution of the average power in CW laer or pule energy in puled laer along the cavity i one of the important characteritic of ytem operation. Spatial dynamic of the energy/average power in a typical laer ytem i defined by the effect of gain provided by an amplifying medium; cavity loe that include both ditributed and point loe, and effect of uration of gain and lo. Note that in general, gain and lo can be frequency dependent either due to the pectral dependence of the gain/lo or a a reult of optical filtering. Of particular interet are mathematical model decribing high power mode-locked laer ytem that preent example of nonlinear diipative ytem in which table tructure (diipative oliton) emerge a a reult of the balance between effect of gain/lo, diperion and nonlinearity. vidently, due to complex nature of uch ytem, there i a variety of poible mechanim of formation of table pule. Note that typically, in many practical ytem, the effect of the pectral dependence of the gain i mall compared to other effect and doe not play a critical role in the generation of table pule. The effect of the urable aborber, on the contrary, i important for elf-tarting of the mode-locking. However, depite being of critical importance for generation of coherent tructure from noie, in ome ytem and operational regime, the urated lo i not effectively involved in tabilization of pule parameter at high energie. In thi work we conider an energy balance equation in ytem with diperion, intantaneou nonlinearity (e.g. Kerr-type nonlinearity), urated gain and non-urated lo. The derived analytical reult provide ueful inight into the evolution of the energy along the cavity. However, we would like to tre the limitation of the conidered, relatively imple mathematical model. Obviouly, in practical laer cheme, epecially at high-power, additional phyical contraint uch a e.g. thermal effect, optical damage limit, pumping ource brightne, mode overlap and other have to be taken into account. Note alo, that our reult are directly relevant to the mode-locking and/or Q- witched fibre laer ytem, but the derived theory i rather general and can be applied in a variety of laer application. We conider here the mater model that i often called either the nonlinear Schrödinger equation with diipative term [ 5] or the Ginzburg-Landau equation [6 6]. In the context of laer application thi model decribe the temporal and longitudinal evolution of the lowly varying optical field envelope A(z,t) along the cavity medium (e.g. the active/paive fiber egment) in which the effect of point laer ytem element (e.g. plicing lo) might be included a the correponding ditributed effect: n n () i βn ( ) γ n n= A A g A i + + A A= i( ) A z n! t () Here the diperive term include higher-order diperion expreed through derivative of the propagation contant β(ω) with repect of omega, γ i the nonlinearity parameter, g i the power dependent (urated) gain and i an effective ditributed lo. The gain i urated in the following manner g g g( A) = =, + Pav / P + / P A() t dt, P T, P T. av = = av R = R TR #6 - $5. USD Received 9 May 9; revied Jun 9; accepted Jun 9; publihed 9 Jun 9 (C) 9 OSA 6 July 9 / Vol. 7, No. 4 / OPTICS XPRSS 899

3 Here TR = nl/c i a round trip time, a homogeneou broadening i aumed, g i a mall ignal gain, P i the uration power (that in the context of fibre laer i a product of the uration intenity by the effective mode area) of the ignal power paing through the amplifying medium that urate gain down to half of the mall-ignal gain g. Here uration effect are intentionally preented either through the pule energy or uing the average power, to reflect the fact that the obtained reult can be applied both in the context of CW laer (when it i more appropriate to ue average power) and in puled laer. Note that the model () doe not include term reponible for the action of urable aborber and for in-cavity filtering and in thi ene i le general then the mater equation introduced by Hau [6,7] or the model conidered in [8] baed on uing the o-called ABCD approach []. In modern powerful laer ytem the effect of nonlinearity and diperion per cavity round trip might not be mall reulting in a trong pectral and temporal evolution during pule propagation through the laer reonator. Generation of laer radiation i determined by the complex interaction of different phyical effect decribed by the mater q. (), with diperion and Kerr nonlinearity playing critical role. However, diperive and nonlinear effect in q. () do not impact the energy of the radiation. In other term, without the gain/lo diipative term q. () i of the Hamiltonian type and the total energy i conerved. volution of the pule energy (average power) along the amplifying medium with an effective ditributed lo i governed by the ordinary differential equation that can be applied to both CW and puled laer: d g = () dz + / Note that thi equation ha been well tudied in context of CW laer and the general olution ha been already decribed in the literature [] and often i called a Rigrod analyi [7]. In thi work we reviit thee reult in the context of pecific cavity configuration to derive an analytical expreion linking the output energy/average power and the cavity gain/lo parameter. Note that uch exact energy evolution analyi, in general, cannot be applied in model ued in [6 8]. Once more, it i important to point out that in many laer ytem, the urable aborber i critically important for initiation of generation, but doe not have trong impact on the haping of the generated pule and the main feature of the energy evolution along the cavity. We would like to point out that the obtained reult are not directly applicable in ytem where trong filtering i playing important role in haping the generated laer pule, for intance a in [8]. Note alo, that our reult are motly applied in the ituation when generated radiation propagate in one direction (e.g. ring cavity), however, the derived reult can be alo ued under certain aumption in more general configuration. Without lo of generality, we will refer in what follow to the pule energy, having in mind that imilar reult can be derived for the averaged power in CW laer ytem. Firt of all, it g i een that the aymptotic tationary value of the energy i = [ ]. Integration of aym thi equation yield the trancendent equation expreing energy a a function of the propagation ditance z along the cavity: z ( ) z ( ) [ ( + )] = exp[( g )( z z)], =. (3) g Here z i a conerved quantity (integral of motion) of q. () that i determined by the initial value of the energy at z =. vidently, in the limit q. (3) diverge into the known #6 - $5. USD Received 9 May 9; revied Jun 9; accepted Jun 9; publihed 9 Jun 9 (C) 9 OSA 6 July 9 / Vol. 7, No. 4 / OPTICS XPRSS 9

4 relation: ln + + = g ( z z ) []. The q. (3) allow u to expre energy at any point along the cavity through the accumulated gain and the initial energy: z ( ) z ( ) () () [ ( )] exp[( )( )] ( ) [ ( )]. + = g z z = G z + It i convenient to introduce a function of two variable f ( x) = x [ x] with and x varying within the interval < x <, < <. The function f ( x ) ha an evident limit f ( x) xexp( + x). The invere function f ( x) can be eaily tabulated and ued in numerical imulation in a way imilar to any elementary function. Figure how typical behaviour of the invere function f ( y) with growing argument y for different value of the parameter. It i important to note that the invere function i calculated and tabulated once and forever and hould not be recalculated each time it appear in equation. Fig.. The invere function x f ( y) = for different value of the parameter. volution of the energy along the cavity then can be preented in a compact form: () z ( ) = f [ Gz ( ) f ( )]. Thi expreion can be ued to preent the balance of gain/lo in different configuration of the laer cavity. It i particular ueful for the analyi of the energy mapping problem evolution of the energy after one round trip and radiation hedding at the output coupler. A an example, we conider a Fabry-Perot cavity cheme where the action of the reflector at the + + edge of the cavity i decribed by the tranformation ( L) = R ( L) and () = R (). Introducing the function f ( x) = x [ x] allow u to expre energy mapping after one round trip { + () + ( L) ( L) () + ()} that include the effect of the two #6 - $5. USD Received 9 May 9; revied Jun 9; accepted Jun 9; publihed 9 Jun 9 (C) 9 OSA 6 July 9 / Vol. 7, No. 4 / OPTICS XPRSS 9

5 mirror (e.g. fibre Bragg grating in the cae of fibre laer) with the reflectivitie R, R in a compact form: () () = R f { G f[ R f ( G f( ))]} (4) Here G = G( L) i one pa cavity gain. The trancendent q. (4) i not jut a mathematical trick allowing to preent energy evolution in the apparently cloed way. Thi preentation greatly implifie calculation of the energy dependence on G, R, R,, = / g a a olution of the mapping problem (4). In the limit / << the mapping q. (4) can be reolved analytically leading to the well-known laer reonator formula R R exp[( g ) L] =. For a ring laer configuration = ( R) ( L) and () = R( L), where L i a length of out the ring cavity. In thi cae, one can derive an analytic expreion linking laer output characteritic (energy or average power) with cavity and amplifying medium parameter: out g R = R [ Lg ( ) + ln R] g Lg ( ) + ( g)ln R] g inh{ } inh{ } (5) Thi compact analytical reult connecting the key ytem and generated radiation characteritic can be ued for calibration of parameter. Note that the analytical formula (5) i derived under aumption of unidirectional ring reonator and homogeneou broadening. In the limit the expreion reproduce the well-known formula []: out = [ln R+ gl]. Note that in the cae of a long enough cavity L (that can be practically defined a the reonator length larger than a cale at which the in-cavity energy g reache it aymptotic value the output energy i = [ ] ( R). aym quation (5) can be re-written uing an effective gain parameter, ΔG( L) = L( g ) auming a net gain in the cavity: R exp[ ΔG] (thi, for a given reflectivity retrict interval of allowed Δ G to exp[ ΔG] / R and vice vera), and a lo/mall gain ratio parameter = in form: g R exp[ ΔG( L)] = ( R) = ( R, L, ) (6) out out R exp[ ΔG( L)] quation (5) i the key reult of the work. Thi equation provide an analytic expreion for the output energy/power a a function of laer ytem and amplifying medium parameter. It i een that a large effective gain mean that energy reache an aymptotic value in one tranit, mathematically thi correpond to the limit Δ G >>, then the exponential term can be neglected and the output energy i determined by the aymptotic value. Figure -4 #6 - $5. USD Received 9 May 9; revied Jun 9; accepted Jun 9; publihed 9 Jun 9 (C) 9 OSA 6 July 9 / Vol. 7, No. 4 / OPTICS XPRSS 9

6 illutrate q. (6) preenting the normalized energy / a a function of, ( ΔG Δ GdB [ ] = log ( e ) = Δ G) and R. Δ G Fig.. Contour plot of the normalized energy / in the plane (gain, R); here =.3. Figure preent dependence of the output energy on R and effective gain. Croe of the horizontal line (correponding to fixed reflectivity) with the energy ioline (contant energy) how how much effective gain i needed to reach a certain output energy for a given R. For intance, it i een that for reflectivity R in the range between.8 and, after certain value, further growth of the effective gain doe not increae the output energy. The vertical tangent to the curve of contant output energy determine a minimal effective gain required to reach uch output energy and a reflectivity R that i optimal (provide for laing with minimal effective gain). Vertical line correponding to a fixed effective gain how that the ame output energy can be achieved for two value of R. Fig. 3. Normalized energy / a a function of an effective gain for different R and. #6 - $5. USD Received 9 May 9; revied Jun 9; accepted Jun 9; publihed 9 Jun 9 (C) 9 OSA 6 July 9 / Vol. 7, No. 4 / OPTICS XPRSS 93

7 Fig. 4. Normalized energy / v. the reflectivity R for different gain and. Figure 3 illutrate that in the limit ( Δ G+ ln R) <, out / ( ) ( ΔG + ln R), (blue and green dahed line); while in the oppoite limit Δ G >> (three firt line from the bottom): out / ( R), and thi limit can be alo een in Fig. 4 (yellow dahed and green dot-dahed line). An important reult from Fig. 4 i that there are optimal out-coupling reflectivitie R that provide the maximum output energy for a given effective gain. It i een from Fig. 4 alo that performance i trongly affected by the lo/gain parameter = / g. In particular, the optimal reflectivity R can be hifted from value cloe to a on olid blue line to very mall R a for dahed green and yellow. Therefore, the ratio of non-urated cavity lo to the mall gain (urated at larger energie) i an important phyical parameter that hould be taken into account in the deign of uch laer. Finally, note that a imilar analytical expreion can be derived for puled laer baed on the Fabry-Perot reonator with one highly reflective mirror ( R = ). The approximate expreion for the energy at the location of the mirror for Fabry Perot cavity i then the ame a q. (5) with the following replacement: L L, R R. In concluion, the energy mapping in laer reonator with urated gain and nonurated lo i examined. An analytical expreion for the output energy/average power in term of the gain uration energy, cavity lo and mall ignal gain parameter i derived for a ring cavity configuration. The decribed approach i applicable to a wide range of CW and puled laer ytem and can be ued for reduction of the pace of optimization parameter and/or for calibration of the model parameter againt the meaured ytem characteritic. I would like to thank A. Latkin and A. Siegman for ueful dicuion and to acknowledge upport of the Royal Society and PSRC. #6 - $5. USD Received 9 May 9; revied Jun 9; accepted Jun 9; publihed 9 Jun 9 (C) 9 OSA 6 July 9 / Vol. 7, No. 4 / OPTICS XPRSS 94