Bandwidth Programmable Optical Nyquist Pulse Generation in Passively Mode-Locked Fiber Laser

Transcription

1 Bandwidth Programmable Otical Nyquist Pulse Generation in Passively Mode-Locked Fiber Laser Volume 7, Number 5, October 2015 Sonia Boscolo Christohe Finot Sergei K. Turitsyn DOI: /JPHOT Ó 2015 IEEE

2 Bandwidth Programmable Otical Nyquist Pulse Generation in Passively Mode-Locked Fiber Laser Sonia Boscolo, 1 Christohe Finot, 2 and Sergei K. Turitsyn 1 1 Aston Institute of Photonic Technologies, School of Engineering and Alied Science, Aston University, Birmingham B4 7ET, U.K. 2 Laboratoire Interdiscilinaire Carnot de Bourgogne, UMR 6303 CNRS- Université de Bourgogne/Franche-Comtè, Dijon, France DOI: /JPHOT Ó 2015 IEEE. Translations and content mining are ermitted for academic research only. Personal use is also ermitted, but reublication/redistribution requires IEEE ermission. See htt:// for more information. Manuscrit received August 2, 2015; acceted August 26, Date of ublication Setember 25, 2015; date of current version October 2, This work was suorted in art by the Leverhulme Trust under Grant RPG-278 and in art by the EPSRC Program Grant UNLOC under Grant EP/ J017582/1. Corresonding author: S. Boscolo ( s.a.boscolo@aston.ac.uk). Abstract: We roose and numerically demonstrate a novel simle method to roduce otical Nyquist ulses based on ulse shaing in a assively mode-locked fiber laser with an in-cavity flat-to sectral filter. The roosed scheme takes advantage of the nonlinear in-cavity dynamics of the laser and offers the ossibility to generate high-quality sincshaed ulses with widely tunable bandwidth directly from the laser oscillator. We also show that the use of a filter with a corrective convex rofile relaxes the need for large nonlinear hase accumulation in the cavity by offsetting the concavity of the nonlinearly broadened ulse sectrum. Index Terms: Mode-locked fiber lasers, ulse shaing, frequency filtering, nonlinear fiber otics. 1. Introduction Sinc-shaed Nyquist ulses in the time domain ossess a rectangular sectrum, enabling bandwidth efficient encoding of data and intrinsically satisfying the Nyquist criterion for zero intersymbol interference (see, e.g., [1], [2], and references therein). These temoral and sectral roerties are of key interest for high-caacity otical communication systems and various other alications. For examle, sinc ulses can rovide substantial erformance imrovement to otical samling devices [3] because their waveform corresonds to the ideal interolation function for the erfect restoration of band-limited signals from discrete and noisy data [4]. Furthermore, the sectral features of sinc ulses could enable the imlementation of ideal rectangular microwave hotonics filters [5] with tunable assband rofiles, thus roviding interesting ossibilities for all-otical signal rocessing [6], sectroscoy [7], and light storage [8]. These attractive roerties stimulated a great deal of research activity in the field of otical Nyquist ulse transmission and generation. Various otical methods have been reorted, including sectral reshaing of mode-locked laser [9] or fiber otical arametric amlification umed by arabolic ulses in combination with a hase modulator to comensate the um-induced chir [10]. Additionally, sequences of very high-quality sinc-shaed ulses were roduced by

3 Fig. 1. Schematic of the laser. EDF: erbium-doed fiber; SA: saturable absorber. the direct synthesis of a flat hase-locked frequency comb using two cascaded Mach Zehnder modulators [11]. An extension of this scheme, including an additional nonlinear otical stage based on four-wave mixing, was roosed in [12] to exand the bandwidth of the generated comb beyond the limit imosed by the electronic bandwidth of the modulators. Furthermore, a Nyquist laser that can directly emit an otical Nyquist ulse train was recently demonstrated [13]. In the laser design demonstrated in [13], the generation of Nyquist ulses is made ossible by a combination of arabolic time-domain shaing at the ulse eak with an otical intensity modulator and sectral-domain shaing on the ulse wings with a sectral filter. In general, the concet of an in-cavity sectral ulse shaer has been roved to have great otential for controlling the dynamics and the outut of mode-locked fiber lasers [14] [16], while obviously entailing a more ower efficient technique than ulse shaing imlemented through direct filtering of a laser outut. In this aer, we roose a novel, simle aroach to the design of a Nyquist laser, which relies on nonlinear in-cavity ulse dynamics and ulse shaing by an in-cavity flat-to sectral filter in a assively mode-locked fiber laser. An imroved aroach, based on the use of a filter's rofile with a corrective convex to, is also roosed to comensate for the concavity of the ulse sectrum obtained after nonlinear exansion in the fiber. The need for large nonlinear hase accumulation in the fiber is thereby relaxed. We numerically show the ossibility to achieve sinc-shaed Nyquist ulses of high quality and widely tunable bandwidth with the roosed scheme. 2. Laser Configuration and Numerical Model The cavity configuration considered is a simle ring cavity, as shown in Fig. 1. The laser consists of a 1 m-long segment of erbium-doed fiber (EDF) with normal grou-velocity disersion (GVD), which acts as the gain and nonlinear element of the cavity, followed by a sectral filter, which realizes the ulse shaing, and a saturable absorber (SA) element. Pulse roagation within the fiber section is modeled with a standard modified nonlinear Schrödinger equation for the slowly-varying ulse enveloe i z tt þ j j 2 ¼ i 2 g þ 1 2 tt (1) where 2 ¼ 25 fs 2 =mm is the GVD arameter, and ¼ 0:005 ðw mþ 1 is the coefficient of cubic nonlinearity of the fiber [17]. The dissiative terms on the right-hand side of (1) reresent linear gain, as well as a arabolic aroximation to the gain rofile with the bandwidth corresonding to 40 nm full-width at half-maximum (FWHM) bandwidth. The gain is saturated according to g ¼ g 0 =ð1 þ W =W 0 Þ, where g 0 ¼ 30 db=m is the small-signal gain, W ¼ R dt j j 2 is the ulse energy, and W 0 ¼ 150 J is the gain saturation energy. The SA is given by a monotonically increasing transfer function T ¼ 1 q 0 =½1 þ PðtÞ=P 0 Š, where q 0 ¼ 0:9 is the unsaturated loss, PðtÞ is the instantaneous ulse ower, and P 0 ¼ 150 W is the saturation ower. The filter is modeled by the sectral resonse Hðf Þ¼Rðf Þ ex½i 2;acc ð2f Þ 2 =2Š, where Rðf Þ is a sectral

4 rofile with sharly decaying edges outside its bandwidth and a Gaussian to, given by 8 >< 2 exðf 2 Þ; jf jg 1 Rðf Þ¼ 4 ; jf j¼ 1 >: 0 otherwise (2) where B ¼ 2= is the sectral bandwidth (defined as the width of both edges of the sectrum tail), is the time interval between zero crossings of the corresonding sinc-function-like imulse resonse, and is an adjustable arameter defining the width and concavity of the sectrum to. Note that it is customary to include a non-unit eak amlitude in the frequencydomain descrition of the filter in order to give the imulse resonse in terms of the normalised sinc function. However, in our numerical simulations we alied the normalised filter's resonse to the Fourier transform of the ulse. Note also that aroximating the Gaussian function by the first two terms of its Taylor series, that is, by a quadratic function, in the filter's resonse would yield similar results to those resented hereafter. For the secial value ¼ 0, Rðf Þ is the rectangular sectral rofile associated with the Nyquist sinc-function imulse resonse [1], [9] sin 2t r ðtþ ¼ (3) 2t while >0 ð G 0Þ yields a convex or inverted (concave or conventional) Gaussian to. A variety of filters with very stee edges and adjustable bandwidth are readily available. The corrective Gaussian-to rofile can be rovided by a rogrammable liquid crystal on silicon otical rocessor, the shae and bandwidth of which can be software configured [18]; such devices are already commercially available. Note that otimization of the filter's rofile is relatively easy as it requires adjustment of only one arameter ðþ. An additional arabolic sectral hase filter can be included into the ulse shaer to add a secific amount of GVD 2;acc (in s 2 )tothecavity and, thus, control the net cavity disersion [16]. The outut of the laser is monitored behind a 70% couler at the outut of the ulse shaer. The numerical model is solved with a standard symmetric slit-ste roagation algorithm, and the initial field is a icosecond Gaussian temoral rofile. 3. In-Cavity Nyquist Pulse Shaing The ulse solutions obtained for a flat-to sectral rofile ð ¼ 0Þ alied to the filter with related zero-crossing ulse durations of 2.5 s, 1 s and 0.5 s are shown in Fig. 2, and comared with the solutions obtained with a convex Gaussian-to rofile. The shae correction factor was otimized for each ulse duration, and no in-cavity disersion control was used in these simulations. It is seen that the use of the basic rectangular shae for the filter's sectral resonse already allows us to obtain sinc-shaed ulses at the outut of the laser that coincide well with the theoretical ones described by (3) and feature a sectrum that is fairly close to the ideal rectangular case. When a convex sectral rofile is alied to the filter, the correction of the non erfectly flattened ulse sectrum after the gain fiber by the filter brings about sinc ulses of enhanced quality and with an almost ideal rectangular sectrum at the laser outut for all the ulse durations being considered. An examle of ulse evolution when the basic rectangular filtering method is emloyed is illustrated in Fig. 3 by lots of the FWHM ulse duration and sectral bandwidth as functions of osition in the cavity. It is seen that both the temoral and sectral widths of the ulse increase in the gain fiber as the ulse acquires a ositive (normal) instantaneous frequency shift or chir. The initial slight decrease of the bandwidth originates from the reshaing of the sectrum from a rectangular-like rofile at the entrance of the fiber to a arabolic-like shae near the eak with a transition to a stee decay in the first stage of evolution in the fiber. After this stage, the

5 Fig. 2. Temoral (left) and sectral (right) intensity rofiles of the outut ulse from the laser for different filter bandwidths. (a) B ¼ 400 GHz ð ¼ 2:5 sþ, (b) B ¼ 1THz ð ¼ 1sÞ, and (c) B ¼ 2THzð ¼ 0:5 sþ. The rofiles obtained with a flat-to ð ¼ 0Þ sectral ulse shaer (black) are comared with those obtained with a convex Gaussian-to sectral shaer (blue). The correction factor is ¼ 6:1, 1.1, 0.18 in anels (a) (c), resectively. Also shown are the calculated waveforms according to (3) (red dotted). A laser without disersion control in the cavity is modeled. Fig. 3. Evolution of the FWHM temoral (blue circles) and sectral (red triangles) widths of the ulse along the cavity for a flat-to sectral ulse shaer with B ¼ 2THzð ¼ 0:5 sþ. A laser without disersion control in the cavity is modeled. sectrum broadens significantly, and eventually develos the stee and structured edges characteristic of self-hase modulation (SPM). The filter and SA reverse these changes. The filter comensates for sectral and art of temoral broadening, and cancels the temoral hase accumulation in the fiber. Note that the root-mean-square ulse duration is increased by the filter

6 Fig. 4. Misfit arameter to a sinc shae at the outut of the laser (red circles) and eak nonlinear hase shift (blue squares) versus filter bandwidth B for a flat-to sectral ulse shaer and lasers without (left) and with (right) assive nonlinear roagation in the absence of in-cavity disersion control. owing to the secific ringing feature (or riles) on the ulse wings accomanied by the generated Nyquist ulse. The SA decreases the ulse duration and attenuates the ulse riles. The rincile of filtering a ulse in frequency and time due to the large chir resent has already been exloited to achieve new mode-locking regimes in fiber lasers [19], [20]. In the resent work, the intended ulse shaing is achieved through enhanced sectral filtering of a nonlinearly sectrally broadened ulse in the cavity [16]. Note that unlike the cavity design resented in [16], here, we do not emloy an additional assive nonlinear roagation stage in the cavity. Notwithstanding, the sectral broadening in the gain fiber segment is still sufficient to oerate a convenient reshaing across a broad range of filter bandwidths. The eak nonlinear hase shift NL ¼ R dz P 0 ðzþ (P 0 is the ulse eak ower), which the ulse accumulates in the gain fiber (also known as the B-integral) is aroximately 3 rad in the examle of Fig. 3. This is accommodated by a sectral breathing factor (defined as the ratio of the sectral FWHM at the inut and the outut of the filter) of aroximately 2. Because the nonlinearly broadened sectrum of the ulse at the outut of the fiber is wider than the filter's sectral resonse and fairly flat within the filter bandwidth, it can be sliced without caring for the details of its structure [21]. This is the enabling mechanism for the formation of ulses of the desired temoral shae at the outut of the filter, imressed on the ulse by the filter's imulse resonse. In order to illustrate the flexibility of the roosed scheme in terms of outut sectral bandwidth, we have changed the zero-crossing duration arameter in the filter's imulse resonse over a wide range, and assessed the quality of the obtained ulses with the metric M 2 ¼ R dt ðj j 2 jj 2 Þ 2 = R dt j j 4.Here, is the ulse being evaluated, and is the theoretical sinc function (Eq. (3)) with the same eak amlitude. In each case, we have verified that the sectrum was close to the ideal rectangular one. In the left anel of Fig. 4, we summarize the results obtained with a flat-to sectral rofile alied to the filter and in the absence of in-cavity disersion control. It is seen that the quality of the generated sinc ulses imroves with increasing values of the filter bandwidth, which enable increasingly higher values of the nonlinear hase shift accumulated in the fiber. This confirms that the ulse-shaing mechanism of our laser design requires SPM, and gets stronger with increasing ulse intensity and nonlinear hase shift.

7 Fig. 5. Misfit arameter to a sinc shae at the outut of the laser (red circles) and correction factor (gray oen squares) versus filter bandwidth B for a Gaussian-to sectral ulse shaer and a laser without assive nonlinear roagation in the absence of in-cavity disersion control. The eak nonlinear hase shift accumulated by the ulse, however, saturates to about 3.5 rad in our laser configuration. Above this value, the ulse sectrum after the fiber begins to slit and oscillations begin to aear in its central art. It is seen in the left anel of Fig. 4 that sincshaed ulses with low M values ðm 0:06Þ are obtained over the bandwidth range from a few hundred gigahertz to a few terahertz. At smaller bandwidths, in order to enhance the sectral breathing, and, hence, the filtering rocess, more nonlinear ulse roagation is necessary in the cavity. To this end, one may either use a longer gain fiber segment or incororate a short segment of assive nonlinear fiber into the cavity. In the sirit of rior work on in-cavity sectral enhancement [16], [22], we laced a 20 cm-long segment of highly-nonlinear hotoniccrystal fiber ( 2 ¼ 7fs 2 =mm and nonlinear coefficient nine times larger than that of the gain fiber) after the gain fiber. As the right anel of Fig. 4 shows, high-quality sinc-shaed ulses were indeed obtained over the bandwidth range from a few ten to a few hundred gigahertz in this case. We confirmed in the simulation that a similar erformance was also achieved with 2 m of gain fiber and without assive nonlinear roagation. For larger bandwidths than the uer bounds of the intervals shown in Fig. 4, stable single-ulsing was not ossible. Fig. 5 outlines the results obtained for the same laser configuration as that shown in the left anel of Fig. 4, but using now the enhanced sectral filtering method. Misfit values to a sinc ulse shae below 0.03 are ossible with this strategy over the full bandwidth range being studied. Remarkably, the misfit arameter is reduced by a factor of more than ten (and even reaching a hundred in some cases) with resect to the rectangular filtering aroach for bandwidths between the lower bound of the interval studied (10 GHz) and 2.2 THz. This relaxes the need for large nonlinear hase accumulation or, in other words, large sectral broadening in the fiber section to obtain a fairly flat ulse sectrum within the filter bandwidth. Indeed, a higher degree of concavity of the outut sectrum from the fiber stemming from a lower extent of nonlinear sectral exansion can be offset by a higher value of the correction factor (see Fig. 2). The increase of the misfit arameter with increasing bandwidth after aroximately 1.5 THz is due to the fact that at such large bandwidths, some structure of the nonlinearly broadened ulse sectrum at the fiber outut falls within the filter bandwidth. Such a structure cannot be efficiently cancelled by alying a simle convex function. Consequently, the strength of the corrective sectral shaing aroach increasingly diminishes with increasing bandwidth, as it is evident from the misfit arameter increasingly aroaching the values obtained with the basic shaing aroach. Ultimately, the corrective aroach becomes disensable. However, within such a bandwidth range, the generated sinc ulses with the basic aroach are already of very high quality (see Fig. 4) and, thus, do not comulsorily require corrections. We also studied the effect of net cavity disersion on the quality of the generated Nyquist ulses. The results resented in Fig. 6 show that the tolerance of Nyquist ulse shaing to the cavity disersion increases with increasing ulse duration, while the disersion range where the best sinc-shaed ulses are obtained shifts toward increasingly higher values of normal

8 Fig. 6. Misfit arameter to a sinc shae at the outut of the laser versus net cavity disersion for: a flat-to sectral ulse shaer with B ¼ 2 THz and a cavity without assive nonlinear roagation (red circles), flat-to (blue triangles), and Gaussian-to (blue oen triangles) sectral ulse shaers with B ¼ 1 THz and a cavity without assive nonlinear roagation, a flat-to sectral ulse shaer with B ¼ 300 GHz, and a cavity with assive nonlinear roagation (black squares). The natural disersion of the cavity is shown by a dashed line. disersion when the basic rectangular filtering method is emloyed. Fig. 6 also highlights a tyical scenario of when a corrective Gaussian-to rofile is alied to the filter: When anomalous GVD is added into the cavity, a larger shae correction factor can comensate for the higher degree of concavity of the ulse sectrum at the outut of the fiber to some extent, and so better quality sinc-shaed ulses can be obtained comared to the basic filtering method. On the contrary, with increasing normal disersion in the cavity, the ulse sectrum at the outut of the fiber acquires an increasing convex feature, which cannot be offset by a concave filter's sectral rofile ð G 0Þ. Therefore, the imrovement in ulse quality brought about by the corrective method using small negative values of is minimal. Furthermore, we investigated the influence of higher-order disersion of the gain fiber on the generated sub-icosecond Nyquist ulses. For tyical values of the third-order disersion (TOD) coefficient of the fiber from 50 to 100 fs 3 =mm, we did not observe any areciable effects of the TOD on the ulses. This indicates that the higher order disersion does not limit the oeration or erformance of our Nyquist laser. 4. Conclusion We have numerically shown the ossibility of directly generating sinc-shaed Nyquist ulses of high quality from a assively mode-locked fiber laser incororating a simle flat-to sectral filter in the cavity. Pulse shaing in such a laser occurs through filtering of a sectrally nonlinearly broadened ulse in the cavity. We have also shown that the use of a filter's rofile with a corrective convex to can comensate for the concavity of the nonlinearly broadened ulse sectrum in the fiber, thereby relaxing the need for large nonlinear hase accumulation. In rincile, the corrective sectral shaing aroach resented here could be used to synthesize any kind of dissiative solitons in a mode-locked fiber laser with a very high recision. The filtering rocess being used enables tunability of the outut sectral bandwidth over a wide range, u to a few terahertz. The high flexibility of the bandwidth can be of articular interest for hotonic assisted technologies such as hotonic analog-to-digital conversion or otical samling. It is also noteworthy that the simlicity of the roosed fiber laser design is aealing for imlementation in exeriments. It is tyical in otical communications that the availability of a certain technical solution refocuses research on the feasibility of simler or more cost-efficient versions. From a

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