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2 Chater 12 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime Sorin Miclo, Dan Savatru, Roxana Savatru and on Lancranjan Additional information i available at the end of the chater htt://dx.doi.org/ /61882 Abtract The aim of thi chater i to highlight the role of imulation method a tool for analyi of low and medium average ower fiber laer oerated in aively Q-witched and/or mode-locking regime into the deign of variou alication uch a material microroceing of enor alication. The chater uroe conit in making available to ecialit in the field of laer, electro-otic and even nano-hotonic imroved rocedure for deigning high-accuracy remote enor dedicated to large range of laboratory, indutrial and military alication. The reaon that thi chater deal with aive otical Q-witching and mode-locking technique tailored for fiber laer i the high ercentage of ening device oerating in thi regime. Numerical imulation reult obtained for thi cla of laer emitter can be ued for other tye of laer, uch a otical fiber laer. There are briefly reented the two main mathematical method ued to analyze olid laer ocillator in aive otical Q-witching regime: the couled rate equation aroach and the iterative aroach. The validation of the reented numerical imulation method i done by comarion with exerimental reult. Keyword: Fiber laer, all-fiber aive otical Q-witching, all-fiber mode-locking numerical imulation 1. ntroduction Erbium, ytterbium and ytterbium/erbium co-doed fiber laer oerated in mode-locking regime (ML) are an extremely ueful tool for an increaing number of reearche, medical and indutrial alication. Their main characteritic i their outut coniting of a hae coherent train of very hort ule, le than a icoecond. The alication range of ML-oerated fiber laer an from micro-machining metal [1] all the way to the mot recie frequency meaurement ever made [2]. aed on the many rooal for new technologie that utilize 2016 The Author(). Licenee ntech. Thi chater i ditributed under the term of the Creative Common Attribution Licene (htt://creativecommon.org/licene/by/3.0), which ermit unretricted ue, ditribution, and reroduction in any medium, rovided the original work i roerly cited.

3 276 Fiber Laer mode-locked laer [3, 4], it i clear that thee laer will be an invaluable tool for future technologie. Here, we reent imulation reult obtained in analyzing a articular tye of mode-locked fiber laer, namely thoe uing erbium (Er), ytterbium (Yb) or ytterbium/erbium-doed or codoed ingle-mode (SM) otic fiber a active medium and/or aturable aborber [5-13]. Thee tye of laer are intenively invetigated becaue of their advantage, i.e. low cot, low ower conumtion, long term of ue, robutne, and eae of long-ditance tranmiion (through ingle-mode fiber). The erformed analyi i ointing to a ecial toic of uch laer, namely the tabilization of the reetition frequency of thee laer. nitial demontration of thee laer howed large amount of high frequency noie in thee ytem [14, 15]. We then turned our attention toward invetigation of reduction of frequency noie in thee ytem for deign imrovement, ynchronizing remotely located fiber laer uing thi fat actuator in conjunction with a tabilized fiber link. The erformed imulation of Er, Yb or Yb/Er fiber laer oerated in ML regime ha the coe of develoing a oftware toolbox dedicated to an otimal laer deign for micromachining gla and bare SM otic fiber. 2. Theory The ML technique i baed on hae locking many different frequency mode of a laer cavity. For ractical reaon, a their bulk olid tate counterart fiber laer are ued in many alication which require high eak ower and ule energy [16-23]. There are two mot well-etablihed technique for uling fiber laer: Q-witching and mode locking in either active or aive form [17, 19-21]. ecaue of it in rinciia imlicity, one of the mot ued device for uling fiber laer i the aive Q-witch cell manufactured, in eence, of a aturable aborber material. Q-witching i an effective method to obtain giant hort ule from a laer by oiling the cavity lo eriodically with a modulator inide the reonator cavity [17, 19]. n thi technique, the um deliver contant ower all time, where the energy i tored a accumulated oulation inverion during the OFF time (high lo). During the ON time, the loe are reduced and the accumulated oulation difference i releaed a intene ule of light. Q-Switching allow the generation of ule of mj energy, n duration and few Hz to hundred of khz reetition rate [17-21]. For horter ule in icoecond or ubicoecond range, mode- locking i the main mechanim [21-23]. The hae locking of different frequency mode of a fiber laer cavity imoe the laer to roduce a continuou train of extremely hort ule rather than a continuou wave (CW) of light [16-18]. n rincile, a continuou train of extremely hort ule can be generated from a aively Q-witched laer. t i alo oible that the laer generate a continuou train of extremely hort ule which have amlitude modulated by the aturable aborber and having an enveloe of the laer ule eak imilar to Q-witched ule [21-23]. The difference between thee two cenario of fiber laer emiion obtained under continuou uming of doed otic fiber active media lie in the otical hae of the ule. The mode-locked ule are hae-coherent

4 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / with each other, while the Q-witched ule are not [24-26]. Thi imle fact ha maive conequence regarding the alication of thee two tye of laer. Thi technique induce a fixed hae relationhi between the longitudinal mode of the laer cavity. nterference between thee mode caue the laer light to roduce a train of ule. Locking of mode hae enable a eriodic variation in the laer outut which i table over time, and with eriodicity given by the round tri time of the cavity [27-33]. To undertand the mode-locking roce, even if it aear too baic, it i ueful to begin by invetigating a CW umed fiber laer, which i uoed to have a CW outut in the frequency domain. For a ingle longitudinal mode CW laer, conidered a having a Fabry- Perot cavity and a frequency defined a n = 2 c σl (1) it can be conidered that one reonant mode of the laer cavity overla in frequency with the gain medium. Thu, thi hyothetical fiber laer emit a CW beam with a narrow range of frequencie defined a ( ) = exé ( w + j ) E t E ù 1 ë i t 1 1 û (2) n general, however, the gain medium could overla with everal mode. We can decribe the outut of uch a laer in the time domain a N ( ) = å exé ( w + j σ σ σ ) E t E ù ë i t û (3) 1 where the um i over all of the laing cavity mode, E σ i the amlitude of the σth mode, ω σ i the angular frequency of the σth mode, and ϕ σ i the hae of the σth mode. For the inglemode laer, thi um jut ha one term a given above. we will ee, the hae term lay the key role in the difference between incoherent multimode laing and mode locking. t can be conidered an increaing range of overla between the gain bandwidth and with fiber laer cavity mode. n thi cae, there are N term in Equation (3). The outut of uch a laer deend critically if there i a well-defined hae relationhi between the N mode. f each mode ha a randomly varying hae with reect to the other mode, then a time domain detector on the outut would how u that the laer i emitting a CW beam with a large amount of intenity noie, while a frequency domain detector would how u that the energy wa contained in narrow ike (with lot of intenity noie) aced evenly by the free ectral range (FSR) of the cavity. However, if we can fix the relative hae to a clearly defined value, then the ituation change dramatically. With fixed hae relationhi, the N mode can combine to

5 278 Fiber Laer interfere in uch a way a to contructively interfere at multile of the roundtri time of the cavity, while they detructively interfere elewhere. Thi roce create horter ule a the number of hae-locked mode increae [27-33]. n obviou quetion to be aked concern how thi well-defined relation between fiber laer cavity longitudinal mode i exactly obtained, imlicitly how the hae locking of the longitudinal mode i obtained. The anwer to thi quetion could be obtained from the time domain icture of mode-locking. t i clearly exerimentally etablihed that a mode-locked laer roduce ultra-hort ule at a rate equal to the round tri time of the otical cavity. The exlanation of thi exerimental obervation i that there ha to be ome art of the fiber laer that allow the ule emiion over CW radiation. Thi tatement equate to ay that there i needed of ome element roviding high lo at low intenity (CW radiation) and lower lo at high intenity (uled oeration). Such a device i a aturable aborber. The oerational rincile of atom or molecule a aturable aborber are traightforward: low-intenity light i aborbed by the atom and re-emitted into 4π teradian (i.e. out of the laer cavity), while high-intenity light fully excite the atom and ae mot of it hoton through the medium. The main feature of the aturable aborber i it decreaing lo with increaing intenity. i een immediately, thi behavior can be mimicked with otical rocee that have nothing to do with actual atomic or molecular reonance abortion. t i intereting to note that the hitory of mode-locked laer began not long after the firt demontration of a continuou wave laing in 1960, and it i intimately connected with earch for obtaining ubtance having a aturable abortion. Maiman [34] ruby laer wa created at Hughe Reearch Laboratory in California; the creation of the firt mode-locked laer would occur at ell Laboratorie in New Jerey. n 1964, Hargrove et al. [35] ued an extremely clever acouto-otic technique to rovide a lo modulation in a helium-neon laer cavity, which led to the laer being actively mode locked. n 1965, Mocker and Collin howed that they could achieve tranient locking of the mode of a multimode Q-witched laer uing a aturable Q- witching dye (crytocyanine in methanol) [17]. Since only a few mode were involved in thi roce, the ule width were on the order of ten of nanoecond. Their technique, however, required no active modulator, and thu wa the firt demontration of aive mode locking. The drawback of thi dye wa that it required the laer to be Q-witched in order to aturate and thu the laer emitted mode-locked ule only at the Q-witched interval. The tranient nature of the mode locked ule roved to be roblematic in ractical alication (ultrafat ectrocoy, nonlinear otic, etc.). Thi roblem wa olved in 1972 when en et al. introduced a laer baed on the aturable dye (Rhodamine 6G) that could mode lock continuouly [18]. The ule from thi laer were found to have ule width of only 1.5 icoecond. fter thi demontration, many reearcher effort were done in order to uh the gain bandwidth further with other tye of aturable aborber, including, in the firt tage, different tye of dye, and after that eriod, olid-tate aturable aborber uch a the one made of emiconductor material (SES M) and of crytal or glae doed with different tye of ion uch a Co 2+, Cr 4+, Zn 2+ and other. n the cae of fiber laer, one newly aearing mode-locking technique conit of uing otic fiber doed with Sm or Tm ion or even with the ame tye of ion a the otic fiber active medium.

6 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / Regarding the ML oeration of fiber laer, it i worth to analyze with a tye of aturable aborber known a an effective aturable aborber. t i a ecial cla of aturable aborber, uing rocee other than atomic/molecular abortion. Thee aturable aborber do not have to rely only on actual atomic tranition. Thi mean that the recovery time for the aturable aborber can be much fater than for atomic tranition. Slow aturable aborber can roduce ule with duration le than a icoecond by hortening the leading edge of the ule via aturable abortion and the trailing edge via gain aturation. However, if the atomic tranition of aturable aborber recover fat enough, it can horten both ide of the ule uing the aturable aborber effect which i achieved by exloiting the intenity- deendent index of refraction defined a σ( ) = σ + σ 0 2 (4) where σ 0 i the index of refraction, σ 2 i the nonlinear index coefficient and i the otical intenity. The recovery time for a aturable aborber baed on thi effect i eentially intantaneou becaue non-reonant otical rocee are extremely fat inducing a nonlinear index to reond on the order of a few otical cycle. Two tye of mode locking baed on effective aturable aborber are motly ued and reorted in literature: Kerr len mode locking (KLM) and additive ule mode locking ( PM). KLM i baed on len creation into the gain medium a an effect of the nonlinear index of refraction, len which caue elf-focuing of the beam [36, 37]. KLM i combined with an intracavity aerture, thi effect create a ituation where the cavity refer uled oeration becaue if the laer i in CW oeration, there i a high lo due to the aerture, while in uled oeration the beam focue through the aerture with minimal lo. PM i realized by interference of circulating ule. n the firt realization of PM [38, 39], thi interference wa between ule roagating in two couled cavitie. The main cavity ha the gain medium and an outut couler, while the econdary cavity ha a nonlinear ection, an otical fiber. Pule that are couled to the nonlinear cavity exerience an intenitydeendent hae hift. When thee ule are couled back to the main cavity they can be made to overla with the normal ule in uch a way a to contructively interfere at their eak, while detructively interfering at their wing. Thu, the addition of multile ule reult in ule hortening on every round tri, jut like a real aturable aborber. One ecial tye of PM i baed on nonlinear olarization rotation (P- PM) [39] and i articularly ueful in a fiber laer cavity. The baic idea i that the added ule are not from earate cavitie, but are co-roagating with different olarization. Ellitically olarized ule roagate in a Kerr medium to roduce nonlinear olarization rotation. Exerimentally, thi ituation can be roduced by inerting a quarter-wave late into the fiber cavity, o that linear olarization can be turned into ellitical. The highet intenity art of the ule (i.e. the eak) undergoe a nonlinear hae hift and thu rotate it olarization ome amount. The wing of thi ule, which have low intenity, do not undergo thi hae hift and thu exerience no rotation. quarter-wave late and linear olarizer at the outut of the Kerr medium (fiber) turn the

7 280 Fiber Laer intenity-deendent olarization into an intenity-deendent tranmiion. Thi tye of mode locking can roduce ule width that are cloe to the gain bandwidth limit of Er ( 100 f) [39]. Mode-locked Er or Yb/Er fiber laer have many advantage to be conidered regarding variou alication. One main advantage conit in the fact that an all fiber cavity need no realignment. lo, the comonent needed to build a mode-locked all fiber laer are relatively chea due to their ma roduction in the telecommunication indutry. an examle, a nonlinear olarization mode-locked all fiber laer could be built for an exene of le than 4, 000 USD. For comarion, a tyical olid-tate titanium doed ahire mode-locked laer could be urchaed from a vendor for around 100, 000 USD. While the tyical outut ower of a Ti:Sahire ytem i roughly an order of magnitude larger than that of a mode-locked Er fiber laer, it i traightforward and inexenive to build an Er amlifier that allow the Erbaed ytem to reach average ower level cloe to thoe of the mode locked Ti:Sahire ocillator. Uing a frequency doubling crytal, one can even tranform the 1, 550 nm centered Er laer to Ti:Sahire wavelength around 750 nm. Finally, the relatively mall gain bandwidth of the Er gain medium can eaily be converted into an octave of ectrum uing highly nonlinear fiber. ll of thee factor have layed a art in the raid emergence of fiber laer in the world of ultrafat hyic in the at 10 year. Erbium-doed fiber i articularly ueful over other rare-earth doed fiber (i.e. ytterbium, neodymium, thulium, etc.) due to ilica gla low lo window in the telecommunication C band (conventional band: 1, 530 1, 565 nm). n order to obtain a better undertanding of the oeration of an Er or Yb/Er-doed mode-locked fiber laer, the rate equation aroach i ueful. Thi aroach tart with the Yb and Er ion energy level diagram, chematically reented in Figure 1. The Er ion i a quai-3 level ytem, meaning that although the lowet tate in the laing cheme i not the true ground tate, it i till low energy enough that it ha ome oulation due to thermal excitation. Figure 1. The Yb and Er active ion energy level diagram. The uming of Er ion i accomlihed by either 980 nm or 1, 450 nm light generated by a emiconductor diode laer. n Figure 2, an examle of mode-locked Er or Yb/Er-doed all-fiber

8 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / laer i reented. Here, uming diode are laer uming diode with CW or uledchoed emiion at 976 nm at ower u to 25 W, combiner and WDM are couling the um ower into a double clad Yb-doed fiber ued a active medium, Yb-DCF i Yb-doed double clad fiber active medium with 15 µm diameter core and 15 m length, HR-F G i a high reflectivity (99 %) F G ued a rear laer mirror and OC-F G i a low reflectivity (15 %) F G ued a laer outut couler. Figure 2. Configuration diagram of aively Q-witched all-fiber laer uing an exce of fiber. Regarding the uming efficiency uing 980 nm diode laer radiation, Yb/Er co-doed ha better erformance in comarion with imle Er-doed fiber laer. The role of Yb ion i the one of aborbing the um radiation at nm and due to it intene fluorecence at wavelength 980 nm, a wavelength cloed to Er ion abortion band, to tranfer otical excitation to Er ion. For ingle Er-doed otic fiber active medium, the 980 nm and the 1, 450 nm cheme reult in imilar efficiencie. The trivalent erbium ion, when umed with 980 nm light, i excited to the 4 11/2 tate, which decay to 4 13/2 (ee Figure 1). The decay between 4 11/2 and 4 13/2 i non-radiative (multile honon decay) and occur within a few µ, while the metatable tate ( 4 13/2 ) ha a lifetime of about 10 m. Since the 4 11/2 tate ha uch a hort lifetime, we can make the aroximation that thi highet excited tate ha zero teady-tate oulation (i.e. no oulation accumulate). Thi aroximation reduce the number of articiating energy level to two. We can now write down the relevant rate equation that decribe the number of erbium ion in the uer (N 2 ) and lower (N 1 ) energy level ( ) dn t 1 = A N ( t) + ( N ( t) - N ( t) ) + ( N ( t) - N ( t) e 1 e 2 e 1 a ) dt hn hn (5) ( ) dn t 2 = - A N ( t) + ( N ( t) - N ( t) ) + ( N ( t) - N ( t) a 2 e 1 a 2 e ) dt hn hn (6) where A 21 i the Eintein A coefficient (invere lifetime) for ontaneou emiion, σ a i the cro ection for timulated abortion at the ignal wavelength, σ a i the cro ection for timulated emiion at the ignal wavelength, σ a i the cro ection for timulated abortion at the um wavelength, σ a i the cro ection for timulated emiion at the um wavelength, i the ignal intenity, i the um intenity, hν i the energy of each individual

9 282 Fiber Laer ignal hoton and hν i the energy of each individual um hoton. y dividing the beam intenity by the hoton energy of that beam, we get the total number of hoton aing through a given area (i.e. hoton flux). To achieve laing, we mut have oulation inverion uch that N 2 > N 1. The threhold for thi condition occur when the ion denity in N 2 jut equal N 1. y etting the Equation (5) and (6) equal and olving for the um intenity, we find the threhold intenity for oulation inverion h n = th (7) t ( - a e ) For a um wavelength of 980 nm, thi intenity i roughly 6 kw/cm 2. Since the mode field area of a ingle-mode Er fiber i around 20 µm 2, the um ower needed to achieve inverion i on the order of a few mw. Thi calculation i for a lole cavity, however. Due to loe in fiber lice, the outut couler and loe in the couling of the um diode to the fiber, the actual um ower required for laing threhold i of the order of ten of mw (tyical 980 nm um diode reach average ower beyond 600 mw). t i alo intructive to look at the evolution of the ignal beam a it roagate through the gain (amlifying) ection of the laer cavity. imle differential equation govern the ignal in the reence of a 2-level gain medium d ( z) ( N ( t) N ( t) 2 e 1 a ) ( z) dz = - (8) with the olution ( ) = 0 ex( ) z gl (9) ( ) ( ) é ex ( ) ( ) z = N t - N t l ù 0 ë 2 e 1 a û (10) where 0 i the intenity entering the gain ection, l i the total length of the gain ection and g i the gain defined a ( 2 e 1 a ) ( ) ( ) ( ) g t = N t - N t (11) For our analyi, we will conider the abortion of the ignal beam to be zero, thu

10 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / ( ) = ( ) g t N t (12) 2 e The gain i then deendent only on the denity of excited atom N 2 and the emiion cro ection of the excited Er atom at the ignal wavelength σ a. The emiion cro ection i a contant, thu to determine the gain we only have to find N 2. Uing Equation (8), we have ( ) dn t 2 = - A N ( t) + (- N ( t) ) + ( N ( t) e 1 a ) dt hn hn (13) n the mall ignal limit, the um intenity i much larger than the ignal intenity ( >> ). Uing thi aroximation along with the fact that we are analyzing a teady-tate cenario (d/ dt 0), we can ignore the term and et the left-hand ide of Equation (13) equal to zero. Solving for N 2 yield ( )( << ) = ( ( ) a ) N t N t 2 1 A hn 21 (14) ( )( << ) = t ( ( ) a ) N t N t 2 1 hn (15) ( )( << ) = t N t R (16) 2 where τ i the lifetime of the excited tate, defined a t = 1 A 21 (17) and R i excitation rate which i defined a ( ( ) a ) = 1 R N t hn (18) Thi equation how that the denity of excited atom in the mall-ignal limit i imly given by the lifetime of the exited tate (τ) multilied by excitation rate R. Uing the fact that g i defined a

11 284 Fiber Laer ( ) = ( ) g t N t (19) 2 e the mall ignal gain i g 0 i defined by the relation ( ) t ( ) g t = N t R (20) 0 2 e the ignal beam i increaed to higher intenity, however, we mut take into account the term in Equation (13) that involve. Solving for N 2 yield N 2 ( t) ( << ) N 2 = 1 + at (21) nd the large ignal gain i thu ( ) = g t g0 1 + at (22) where at i the aturation intenity defined a = 1 at (23) t e nd finally, the differential change in ignal intenity er length of gain in the trong um regime i ( ) d z g = dz at (24) The icture of the ignal evolution i now comlete. t low ignal level, there i an exonential increae in the number of ignal hoton in the gain medium. However, a the ignal level i increaed further, the gain begin to aturate and aymtotically aroache a value defined by ( )» = g t g R (25) at 0

12 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / Thu, a exected, at high ignal level, the ignal intenity increae linearly with the um intenity. The fundamental characteritic of laing, namely, the mall-ignal gain and the gain aturation have now been covered. t i worth to mention an imortant aect related to the mode-locking theory, namely, the frequency comb. With the advent of the frequency comb [40] in the late 1990, mode-locked laer, including the fiber one, began to receive much attention concerning the frequency metrology alication. The frequency comb aear a a imultaneou olution for two imortant uroe earated by a vat ga to be achieved uing mode-locking laer, eecially fiber laer. Thee two uroe are: on one ide, the field of reciion meaurement imoe creation of actuated laer that would have the narrowet oible ectral linewidth and, on the other ide, the field of ultrafat ectrocoy wa mainly intereted in creating extremely hort time domain burt of electric field, which necearily require that the ule have a large ectral bandwidth. Thee two goal, which eem to be in direct ooition of each other, can be achieved imultaneouly with a frequency comb. wa mentioned, the frequency comb i baed on mode-locked laer [40-42]. n fact, modelocking and frequency comb are definition ued interchangeably. Thi i not quite right, however, ince technically a frequency comb really refer to a mode-locked laer that ha been carrier-enveloe hae tabilized. The frequency comb can be undertood by uing and by combining it decrition in time and frequency domain [40-42]. The time domain outut of the laer can be viewed a the multilication of the fat electric field ocillation and an enveloe function, rereenting a modulation of electric field amlitude by the enveloe function. The ultimate limit on the width of thi enveloe would be an enveloe that encomae only 1 cycle of the electric field, in other word, correonding to one round tri along laer cavity. t can be hown that the enveloe travel at a eed known a the grou velocity v g defined a v g c = dσ σ - l d l (26) while the electric field fat ocillation travel at the hae velocity v defined a v c = σ (27) Thee two velocitie are, in general, not equal and thu lead to a walk-off or liage between the two entitie, known a carrier-enveloe offet hae. Uing the hift theorem of Fourier tranform [42], we ee that the Fourier tranform turn thi time domain hae li into a frequency offet, f 0. Thu, the otical frequencie of the comb can be defined in term of two frequencie a

13 286 Fiber Laer n σ = σ f re + f 0 (28) where ν σ i the otical frequency of the σth comb mode and f re i the reetition frequency of the laer, which i related to the otical length of it cavity. Clearly, a random variation of the offet frequency would mear out the comb in frequency ace and make it uele for any ort of reciion meaurement. n analogy to thi ort of meaurement would be like trying to meaure the length of omething with a ruler that i alway moving back and forth lightly. Thu, it i clear that to do any ort of reciion meaurement with a mode-locked laer, we need to tabilize thi offet frequency (and thereby roduce a frequency comb). The firt technique that achieved the ability to meaure (and thu tabilize) f 0 relied on the ocalled f-2f interferometer, a comlicated exerimental etu which i quite a technical achievement by itelf. Thi technique i baed on a imle maniulation of Equation (28). n thi cheme, light from a Ti:ahire laer wa ent through a highly nonlinear fiber with low net dierion to broaden the bandwidth of the ule to an octave [40]. The octave anning ule were then couled into an interferometer where in one arm the light wa aed through a econd harmonic crytal and underwent um-frequency-generation (SFG). The two beam were then recombined on a beam-litter, ent through an otical filter, and detected onto a hotodetector to roduce a heterodyne beat at f 0. The octave anning ule bandwidth enure that we have otical frequencie reent in a range from ν σ to ν 2σ, while the econd harmonic arm convert the ν σ light to ν 2σ light via SFG. Filtering out the highet frequencie with the otical filter, and uing the frequency comb equation, we can thu write the frequencie reent in the two arm a n = 2σ f + 2n f = 2σ f + 2 f (29) 2σ re σ 0 re 0 Once thee two beam form a heterodyne beat on the hotodetector, we can take the difference frequency which i ( ) ( ) 2n - n = 2σ f + 2 f - 2σ f + f = f (30) σ 2σ re 0 re 0 The firt demontration of thi method [40] oened the door for exeriment involving the frequency comb. Preciion metrology benefited dramatically from the comact all-in-one nature of the frequency comb, while new technique uch a broadband cavity-ring down ectrocoy [43-44] have been develoed baed on the comb. nother theoretical iue of mode-locked fiber laer concern the conequence of nonlinear effect roduced in the otic fiber during ultrahort light ule roagation. t can be noticed that Equation (5 25) decribe what hould be defined a the energetic art of the mode-locked ultrahort laer roagation henomena. n Equation (5 25), the cycle um abortion

14 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / oulation inverion light energy at the laer wavelength i analyzed. Equation (26 29) are helful for defining the aect related to the reetition frequency of ultrahort light ule. t i to be oberved that becaue of the fiber otic guiding effect, the ultrahort light ule roagation, even with energie on the nj cale and full width half meaure time duration of or f i haening through core tranvere area, meaning extremely large light intenity. Due to the quare variation law of light intenity veru it electric field amlitude, it can be concluded that nonlinear effect can be roduced. Quantitatively, all thee qualitative comment can be develoed tarting from the ytem of Maxwell equation decribing electromagnetic ule roagation through otic fiber. n the ytem of Maxwell equation, the olarization vector of the dielectric roagation medium can be lit into two term, one linear and the other correonding to it nonlinear variation. 2 A A i A 2 + b + b = g i A A z t 2 t (31) where A rereent the roagating electromagnetic field otential, the β 1 -term i reonible for the grou velocity (with the tranformation T=t-β 1 z which tranform to the moving frame of the ule), β 2 i often called the grou-velocity dierion (GVD) arameter. Thi equation i an examle of the nonlinear Schrödinger equation (NLSE). 3. Simulation Reult reviouly mentioned, the imulation of the Er- or Yb/Er-doed all-fiber laer wa erformed aiming to the develoment of a et of toolbox ueful for evaluation of aively Q- witched and/or mode-locked laer dedicated to gla and otic fiber micro-roceing. For accomlihing thi tak, ene reviou exerience in thi field contitute an advantage [45-47]. The imulation crit can be groued into two et: the firt one develoed on the bae of Equation (1-3) and decribing the arameter of a ML fiber laer at the firt level and the econd, the more comlex one, uing Equation (5-22) which treat the roagation of ultrahort laer ule through laer reonator. The econd et i an attemt to olve nonlinear Schroedinger equation. n Figure 3, 4, 5 and 6 are reented reult obtained in invetigation of the noie factor role in definition of mode-locked laer ule emitted by an Yb/Er fiber laer a chematically reented in Figure 2, uing the imulation code of the firt et. CW um ower of 55 W wa conidered. The imulated mode-locked Yb/Er laer i comoed of a double clad fiber having a length in exce (9 m). The value of noie factor, d, i increaed from zero u to 1.5. Figure 3 to 6 rereented the overlaing between gain bandwidth and fiber laer reonator mode. n Figure 3 d i conidered a zero and the mode-locked laer ule are clearly defined.

15 288 Fiber Laer Figure 3. Laer ule intenity veru time imulated for an Yb/Er fiber laer with 9 m length, exce of fiber, conidering a noie factor d = 0. Figure 4. Laer ule intenity veru time imulated for an Yb/Er fiber laer with 9 m length, exce of fiber, conidering a noie factor d = 0.05.

16 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / n Figure 4, a mall increae of d value, u to 0.05 wa conidered and ome noie aearing on the wing of the mode-locked ule can be oberved. n Figure 5, the d factor ha a value of 0.5 and the noie on the wing of mode-locked ule i more evident. n Figure 6, the noie factor i increaed u to 1.5 and it effect on the wing of mode-locked ule are viible. Thi roce create horter ule a the number of hae-locked mode increae. n Figure 7, 8 and 9, the imulation reult are obtained uing the econd et of code in the cae of the ame reviouly defined mode-locked Yb/Er all-fiber laer. n Figure 7, a can be oberved, the mode-locked laer ule are reented a analyzed in time domain. n Figure 8 reult of the ame analyi obtained in defining the fiber laer ule ectrum are reented conidering frequency a inut are reented. n Figure 9, the imulated ule ectrum i reented but conidering wavelength a the argument. Figure 5. Laer ule intenity veru time imulated for an Yb/Er fiber laer with 9 m length, exce of fiber, conidering a noie factor d = 0.5. n each Figure (7 to 9), the imulation wa erformed for three moment: t 0 the initial moment, t t r and t t r, where t r i the laer cavity round tri time. Simulation of the roagation equation (nonlinear Schrödinger equation) reented in Figure 7, 8 and 9 are helful for the undertanding of ule roagation in otical fiber. Eecially chromatic dierion, nonlinear effect and their interlay, leading to hyical henomena like oliton, can be invetigated with thee imulation. The oibility to take a characterized ule and to let it roagate through variou fiber (a long a the fiber arameter are known) i helful for the undertanding of the ule comreion after the amlifier.

17 290 Fiber Laer Figure 6. Laer ule intenity veru time imulated for an Yb/Er fiber laer with 9 m length, exce of fiber, conidering a noie factor d = 1.5. Figure 7. Laer ule intenity analyzed in time domain in the cae of an Yb/Er fiber laer with 9 m length, exce of fiber.

18 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / Figure 8. Laer ule ectrum imulated for an Yb/Er fiber laer with 9 m length, exce of fiber, conidering frequency a inut. Figure 9. Laer ule ectrum imulated for an Yb/Er fiber laer with 9 m length, exce of fiber, conidering wavelength a inut.

19 292 Fiber Laer The imulation alo made it oible to invetigate the ectral broadening of an ultrahort laer ule in a highly nonlinear fiber. The reult how that it i theoretically oible to create an octave anning uercontinuum with the ule from our laer ytem. The imulation make oible the deign of a meauring ytem for the carrier-enveloe frequency baed on a f-2f interferometer. t i oible to invetigate the ectrum that barely an a factor of two into an attemt to detect a carrier-enveloe et frequency beat above the noie. The erformed imulation ha a a future objective, the contruction of mode-locked Er allfiber laer uing a ecial technique, the olarization additive-ule mode locking (P- PM) rocedure. t i baed on the fact that due to the nonlinear refraction of the otic fiber, different intenitie ee a different index of refraction. For ellitically olarized radiation, the net reult i olarization rotation. The invetigated P- PM mechanim work like thi: a ule with a trongly ellitic olarization i ent into a Kerr-medium; deendent on the intenity there will be more or le olarization rotation; combined with a olarizer which only tranmit the rotated art (higher intenity), thi act a a "ulehortener". The mode-locking element are located in the free-ace ath of the fiber laer. The olarizer i rereented by a olarizing beam-cube and a Faraday-iolator or by an otic fiber olarization controller. The quarter-wave late (QWP) right, or the otic fiber which have thi function, make the olarization ellitical; the other two wave late change the olarization to maximize the tranmiion at uled oeration. t can be noticed that the mode-locking i elf-tarting, meaning that the ule build u from initial CW fluctuation, jut a it i the cae of an exce Er fiber laer (ee Figure 2) and i analyzed in Equation (1-3). 4. Concluion The main uroe of thi chater i to reent a oible develoment of imulation code alicable for Er or Yb/Er all-fiber laer oerated in mode-locking regime with gla and fiber otic micro-machining alication. n thi ene, the baic theoretical notion are defined in Section 2: Theory and examle of imulation code ue could be oberved in Section 3: Simulation Reult. fairly good agreement between the imulated data and the imilar one reorted in literature can be noticed. Author detail Sorin Miclo *, Dan Savatru, Roxana Savatru and on Lancranjan * ddre all correondence to: miclo@inoe.ro National ntitute R&D of Otoelectronic - NOE, Magurele, lfov, Romania

20 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / Reference [1] Otendorf, Kamlage G, Chichkov. Precie dee drilling of metal by femtoecond laer ule. RKEN Review: Focued on Laer Preciion Microfabrication. 2003;50:87-9. [2] Oky W, et al. Single-atom otical clock with high accuracy. Phy Rev Lett 2006;97: [3] Ye J. bolute meaurement of a long, arbitrary ditance to le than an otical fringe. Ot Lett 2004;29: [4] Schibli T. Comb for dark energy. Nat Photonic 2008;2: [5] Tai T-Y. et al. ll-fiber aively Q-witched erbium laer uing mimatch of mode field area and a aturable-amlifier um witch. Ot Lett 2009;34: [6] Wu, Chu PL. Fat otical witching in Sm 3+ -doed fiber. EEE Phot Technol Lett 1996;8: [7] Fotiadi, et al. ll-fiber aively Q-witched Ytterbium laer. Proc CLEO/Euroe 29, CJ2-3, [8] Tai T-Y, et al. Saturable aborber Q-and gain-witched all-yb 3+ all-fiber laer at 976 and 1064 nm. Ot. Exr 2010;18: [9] Moore S W et al. 400 µj 79 n amlified ule from a Q-witched fiber laer uing an Yb 3+ -doed fiber aturable aborber. Ot. Exr 2012;20: [10] Soh D S, ion SE, et al. () High-ower all-fiber aively Q-witched laer uing a doed fiber a a aturable aborber: numerical imulation. Ot. Lett 2011;36: [11] Fotiadi, et al. () Dynamic of ll-fiber Self-Q-witched Ytterbium/Samarium Laer. Proc CLEO/QELS/Ph ST 15, CMC4, [12] Fotiadi, et al. ll-fiber coherent combining of Er-doed amlifier via refractive index control in Yb-doed fiber by two-wavelength otical ignal. Proc CLEO/ Euroe, CJ3-5, [13] Huang JY, et al. High-ower 10-GHz elf-mode-locked Nd:LuVO4 laer. l Ot 2008;47: [14] Hong F, Minohima K, Onae, et al. road-ectrum frequency comb generation and carrier-enveloe offet frequency meaurement by econd-harmonic generation of a mode-locked fiber laer. Ot Lett 2003;28: [15] Hudon DD, Holman KW, et al. Mode-locked fiber laer frequency-controlled with an intracavity electro-otic modulator. Ot Lett 2005;30:

21 294 Fiber Laer [16] Hargrove LE, Fork RL, Pollack M. Locking of HeNe laer mode induced by ynchronou intracavity modulation. l Phy Lett 1964;5:4-5. [17] Mocker HW, Collin RJ. Mode cometition and elf-locking effect in a Q-witched ruby laer. l Phy Lett 1965;7: [18] en EP, Shank CV. Diene, Paive mode locking of the CW dye laer. l Phy Lett 1972;21: [19] R(diger P. Encycloedia of Laer Phyic and Technology, 1ed. John Wiley & Son, [20] Digonnet MJF. Rare-earth-doed fiber laer and amlifier, 2d ed. Marcel Dekker, 2001, vol. 25. [21] Svelto O, Hanna DC. Princile of Laer. Sringer, [22] Saleh E, Teich MC, Mater R. Fundamental of Photonic, Second Edition. John Wiley & Son, [23] Wang Y, Xu C-Q. () ctively Q-witched fiber laer: Switching dynamic and nonlinear rocee. Prog Quantum Electronic. 2007;31: [24] Schreiber T, Nielen CK, et al. Microjoule-level all-olarization-maintaining femtoecond fiber ource. Ot Lett 2006;31: [25] Seung C, Song H, Gee S, Dug YK. Self-tarting aive mode-locked ytterbium fiber laer with variable ule width. Proc. SPE - Fiber Laer V: Technology, Sytem, and lication. 2010;7580:75802C. [26] Ortac, Plotner M, et al. Exerimental and numerical tudy of ule dynamic in oitive net-cavity dierion mode locked Yb-doed fiber laer. Otic Ex 2007;15: [27] Jang GH, Yoon TH. Environmentally-Stable ll-normal-dierion Picoecond Ybdoed Fiber Laer with an chromatic Quarter-wave-late. Laer Phy 2010;20: [28] Lian FQ, Fan ZW, et al. Ytterbium doed all-fiber-ath all-normal dierion modelocked laer baed on emiconductor aturable mirror. Laer Phy 2011;21: [29] Turchinovich D, Liu X, Laeggaard J. () Monolithic all-pm femtoecond Yb-fiber laer tabilized with a narrow-band fiber ragg grating and ule-comreed in a hollow-core hotonic crytal fiber. Otic Exre. 2008;16: [30] Tian X, Tang M, et al. High-energy wave-breaking-free ule from all-fiber modelocked laer ytem. Otic Ex 2009;17: [31] Song R, Chen H, et al. SES M aively mode-locked fiber laer with a long cavity including a band a filter. J Otic. 2011;13:

22 Numerical Simulation of Fiber Laer Oerated in Paively Q-Switched and Mode-Locked Regime htt://dx.doi.org/ / [32] Liu J, Xu J, Wang P. High Reetition-Rate Narrow andwidth SES M Mode-Locked Yb-Doed Fiber Laer. EEE Photonic Technol Lett 2012;24: [33] Prochnow O, Ruehl, et al. ll-fiber imilariton laer at 1 µm without dierion comenation. Otic Ex 2007;15: [34] Maiman TH. Stimulated otical radiation in ruby. Nature. 1960;187: [35] Hargrove LE, Fork RL, Pollack M. Locking of HeNe laer mode induced by ynchronou intracavity modulation. l Phy Lett 1964;5:4-5. [36] Sence DE, Kean PN, Sibbett W. 60-fec ule generation from a elf-mode-locked Ti:ahire laer. Ot Lett 1991;16:42-4. [37] Ell R, et al. Generation of 5-f ule and octave-anning ectra directly from a Ti:ahire laer. Ot Lett 2001;26: [38] Mollenauer LF, Stolen RH. The oliton laer. Ot Lett 1984;9:13-5. [39] Stolen RH. Nonlinearity in fiber tranmiion. Proc EEE 1980;68: [40] Jone DJ, Diddam S, et al. Carrier-enveloe hae control of femtoecond modelocked laer and direct otical frequency ynthei. Science. 2000;288: [41] Cundiff S, Ye J, Hall J. Otical frequency ynthei baed on mode-locked laer. Rev Sci ntrum 2001;72: [42] Walker R, Udem T, et al. Frequency deendence of the fixed oint in a fluctuating frequency comb. l Phy 2007;89: [43] Cundiff S, Ye J. Colloquium: Femtoecond otical frequency comb. Rev Mod Phy 2003;75: [44] Everon KM, Well JS, et al. ccurate frequencie of molecular tranition ued in laer tabilization: the 3.39-µm tranition in CH4 and the and µm tranition in CO 2. l Phy Lett 1973;22: [45] Lancranjan, Miclo S, Savatru D. Numerical imulation of a DF -fiber laer enor (). J Ot dv Mat 2010;12: [46] Lancranjan, Savatru D, et al. nalyi of a aively q-witched Nd:Y G lab laer ocillator/amlifier ytem. Proc SPE 2012;8547: [47] Savatru D, Vlae, et al. Numerical imulation of laer technique for art conervation Part Fiber laer analyi. UP Sci ull-ser. 2011;73:

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