SESAMs，composed of a Saturable Absorber（SA）and a Bragg reflector，are the new mode-locked devices developed by the procedure of Metal-Organic Chemical Vapor Deposition（MOCVD）. Compared with the mode-locking technology with a SA^［1］，a Nonlinear Polarization Rotation（NPR）^［2］，and a Nonlinear Amplifying Loop Mirror（NALM）^［3］，which are seriously affected by the environment and have more complex structures，the SESAM mode-locking technology has even higher stability and controllability and is easier to practically apply. Such regime plays an important role in the construction of picosecond fiber lasers.

Since 1992，Keller et al.，first realized the passive mode-locking of solid-state lasers by using a Fabry-Perot etalon SA，SESAMs have undergone several innovations^［4］. At present，SESAMs have been widely used in the realization of pulse fiber lasers^［5，6］ and pulse solid-state lasers^［7-8］. Since then，many researchers have paid their attentions in such lasers^［9，10］. In the mode-locked fiber laser based on a SESAM，ring cavity^［11］ and linear cavity^［12］ are commonly used. In 2008，Ori Katz et al. built an all-fiber picosecond pulsed fiber laser using a SESAM^［13］. The dispersion in the cavity was compensated by a chirped fiber grating，and the polarization state in the cavity was controlled by a polarization controller. Finally，a 1.06 μm mode-locked pulse laser with the pulse width of 3.8 ps was obtained，and the spectrum shape of outputted pulses exhibited a non-Gaussian distribution. Such a laser was amplified as a seed source，which had the great influence on the spectral characteristics of the amplified pulse. In 2016，Wang Zi-Wei et al.，developed a high peak-power picosecond photonic crystal fiber amplifier using a SESAM mode-locked fiber laser as the seed source^［14］. After multi-stage amplifications，a high-powered pulse laser was obtained with the pulse width of 30 ps and the peak power of 2.94 MW. The output characteristics of the seed light are not well satisfactory，which brings about obvious distortion in amplified pulses. In 2018，Sun Jiang et al. built an all-polarization-maintaining ytterbium-doped fiber laser with a linear cavity based on the SESAM mode-locking^［15］. The chirped fiber Bragg grating（CFBG）was used to compensate dispersion，and the all-polarization-maintaining structure was used to control the polarization state in the cavity，and the fiber focusing device was used to spatially couple the SESAM. Finally，the mode-locked pulse laser was obtained with the center wavelength and pulse width of 1.03 μm and 4.26 ps，respectively. The frequency domain output characteristics of the pulse are not satisfactory，and its spatial coupling SESAM mode is harmful to the integrity of such a fiber laser. Its all-polarization-maintaining fiber structure greatly increases the price of the system. In addition，there are many studies on fiber amplifiers using SESAM-based mode-locked fiber lasers as the seed source. However，most of the SESAM-based mode-locked fiber lasers are somewhat complicated and expensive. Additionally，the improvement of outputted characteristics of mode-locked pulses has not been mentioned in time domain and frequency domain. The frequency-domain output characteristics of a mode-locked fiber laser are the key factors to obtain the stable mode-locked pulse output. The higher the edge smoothness of the outputted spectrum is，and the closer the shape is to the ideal line，the more stable the output power of a mode-locked fiber laser is，and the more difficult the losing-lock becomes，the easier the amplified pulses with better output characteristics in the subsequent amplification are obtained. During constructing an ideal seed source for high-power pulsed fiber lasers，it is necessary to optimize the overall structure of a SESAM mode-locked fiber laser under the premise of ensuring the stable output，and to maintain the satisfied output characteristics in both time domain and frequency domain.

In order to make the correlation characteristics of outputted pulses for a mode-locked fiber laser with the compact linear cavity more ideal，and to further reduce the development cost，we build a SESAM-based passively mode-locked fiber laser with the non-polarization maintaining linear cavity by the use of the theoretical simulation in this paper. The cavity length can be further shortened by using the ytterbium-doped gain fiber with the relatively high absorption coefficient，where any polarization controllers have not been employed. Without compensating the dispersion in the cavity，we obtained the stable mode-locked pulse output with the center wavelength of 1.06 μm，the pulse width of less than 12 ps，the 3 dB spectral width of 0.32 nm，the repetition rate of 37 MHz，and the output power of 2 mW，respectively. The spectral edge of the pulse is smooth，and the spectral shape is very close to the Gaussian distribution. The similar research reports have been proved to be very rare. This paper might put forward to a valuable route for the commercialization of the next-generation SESAM mode-locked fiber lasers.

The structure of a linear-cavity fiber laser using the SESAM-based mode-locking is shown in Fig. 1. An Yb-doped gain fiber with high absorption coefficient was used as the lasing element. A SESAM is used as both the mode-locking device and the high reflector. A FBG is used as the wavelength selection device. The resonant cavity is composed of the SESAM and FBG. A 975 nm semiconductor laser is used as the pump source. The output pump light is directly coupled to the resonant cavity by a Wavelength Division Multiplexer（WDM）. The mode-locked laser pulses are outputted from the FBG end，which was acted an output coupler.

When the pulse is transmitted in a single-mode fiber，the relationship is followed by the nonlinear Schrodinger equation as expressed by

where A is the pulse intensity，T is time，β₂ is the second-order group velocity dispersion，β₃ is the third-order group velocity dispersion，γ is the nonlinear coefficient，α is the absorption coefficient，respectively. When the pulse is transmitted in the Yb-doped fiber，the variation of refractive index n₂ is specified by the electric polarization rate of doped ions χd：

where E is energy. After considering the bandwidth limitation caused by the gain，the group velocity dispersion，β₂^eff，of the gain fiber can be expressed by

where T₂ is the dipole relaxation time，g is the gain coefficient following

where E_p is the instantaneous pulse energy and E_sat is the gain saturation energy，respectively.

Therefore，when the pulse is transmitted in the fiber doped with rare earth ions，the physical relationship can be given by the following Ginzburg-Landau equation：

Two terms on the right side of the equation characterize the nonlinear effect and gain-loss relationship，respectively.

The radius of mode field is a parameter to describe the energy transmission concentration of a single-mode fiber，which is important in estimating the fiber loss and dispersion. Figure 2 is diagramed the output intensity of mode-locked pulses with different mode field radii of a single-mode fiber with 2.2，2.95，3.1，3.25，and 3.75 μm，respectively. Figure 3 illustrates the change trend of the pulse width and spectral width for different mode field radii. As the mode field radius of the single-mode fiber increases from 2.2 to 3.75 μm，the Full Width at Half Maximum（FWHM）of the output mode-locked pulse broadens from 9.39 to 12.88 ps，and the FWHM in spectral merely decreases from 5.69 to 4.83 nm，respectively. Figures 2 and 3 show that one can obtain the relatively narrow pulse width using a single mode fiber with the small mode field radius. The single-mode fiber with a larger mode field radius has a larger core diameter，which will reduce the laser power density and weaken the nonlinear effect in the cavity，resulting in the gradual narrowing of the spectrum. According to the principle of time bandwidth product，the increase of the pulse width and the decrease of the pulse amplitude can be observed in time domain. By considering the pulse density drops for a big cross section，it is judicious in selecting a single mode fiber with the small core in which the nonlinear effect is not serious.

In Fig. 4，the influence of the length of a gain fiber on the output pulse characteristics is shown. The absorption coefficient of the gain fiber is 1200 dB/m. The dispersion coefficient of the gain fiber is -35.905 ps/nm*km. When the selected length of a gain fiber is increased from 0.11 to 0.17 m，the spectral width increases gradually from 2.84 to 8.36 nm because the nonlinear effect gradually enhances. Since the pump energy can be absorbed more sufficiently for a gain fiber as long as 0.17 m，the pulse width reduces to 7.87 ps. Figure 5 shows the variation trend of the pulse width and spectral width with seven kinds of gain fiber lengths. If the length of a gain fiber is too short，the pulse width becomes too big and the mode-locking even cannot maintain. If the length of a gain fiber is too long，the nonlinear effect becomes more serious and is harmful to the amplification process. Therefore，we select the length region of the gain fiber as 0.12-0.14 m，which is proper to achieve the suitable pulse width and spectral width at the same time.

The reflectance of a FBG influences the energy returned to the resonant cavity and outputted pulse characteristics. The output features of mode-locked pulses are shown in Fig. 6 with the FBG reflectance of 20%，50%，70%，80%，and 90%. When the reflectance of a FBG is increased from 20 to 90%，the spectral width gradually decreases from 10.02 to 5.77 nm. The gradual narrowing of the spectral width will further reduce the pulse broadening caused by dispersion. One can see from Fig. 6 that the even narrower spectral width and pulse width can be obtained by appropriately increasing the reflectance of a FBG. The variation trend of the pulse width and spectral width are also investigated for 14 kinds of reflectance as shown in Fig. 7. The simulation results show that when the reflectance is 80%，the pulse and spectral widths are 9.65 ps and 7.24 nm，respectively. When the reflectance is raised to 90%，the pulse and spectral widths are 9.39 ps and 5.77 nm，respectively. Thus，we select the reflectance region of the FBG as 80%-90%，which is appropriate to simultaneously achieve the short pulse width and suitable spectral width.

The greater the modulation depth of a SESAM is，the stronger the ability of a SESAM to absorb the weak signal becomes，and the easier the realization of mode-locked self-starting is. Figure 8 shows the output of mode-locked pulses when the modulation depth of a SESAM is assumed to be 1%，10%，20%，35%，and 45% according to the actual parameters from the SESAM maker. In Fig. 9，the trend of the pulse width and spectral width with the different SESAM modulation depths. As the modulation depth of a SESAM is increased from 1 to 45%，the pulse width is decreased from 17.59 to 8.09 ps，and the spectral width is broadened from 5.88 to 6.5 nm. From the plot，one can observe that within a certain range，the larger the modulation depth of a SESAM，the narrower the pulse width becomes. Actually，a large modulation depth will bring the great unsaturated loss，and the small modulation depth will increase the difficulty of mode-locked self-starting. In this study，considering the balance of the pulse width and spectral width，we think the suitable region of the modulation depth of a SESAM is of 15% - 25%.

Unsaturated loss is one kind of depletions which exists even in the saturated state of a SESAM. Insufficient reflectivity，rough surface，impurities，etc.，of a SESAM can all lead to unsaturated loss. In Fig. 10，the output characteristics in both the time domain and frequency domain are given with the unsaturated loss of a SESAM of 0.7%，10%，15%，20%，and 25%. It is obvious that the pulse width varies a little and the spectral width is almost unchanged for different unsaturated losses.

In Fig. 11，the variation trend of the pulse width and spectral width is plotted as the function of unsaturated loss of a SESAM. As the unsaturated loss of a SESAM is increased from 0.7% to 25%，the pulse width decreases from 9.85 ps to 8.49 ps，and the spectral width is narrowed from 6.53 nm to 6.07 nm. Although the pulse width and spectral width decrease slightly with the increase of unsaturated loss of an SESAM，the outputted power will decrease rapidly because of the big unsaturated loss. Therefore，it is better for a user to select a SESAM with low unsaturated loss.

The saturation flux is the photon energy per absorption cross section when the reflectance of a SESAM is 1/e of its modulation depth，which is directly related to the pump power required to achieve mode-locking self-starting. In Fig. 12，the output intensity of mode-locked pulses is shown when the saturation flux of a SESAM is 15，25，50，100，and 130 μJ/cm².

Fig. 13 shows the relationship between the pulse width and spectral width and the SESAM saturation flux. As the saturation flux of a SESAM increases from 15 to 130 μJ/cm²，the pulse width of the mode-locked pulse is obviously broadened from 8.73 to 14.6 ps，and the spectral width is narrowed from 6.95 to 5.03 nm. The broadening in the pulse width and the narrowing in the spectrum correspond to the continuous enhancement of nonlinear effects of a SESAM. It can be seen that the saturation flux has the more obvious effect on the pulse width than on the spectral features. If the saturation flux is too small，the mode-locking self-starting will become somewhat difficult. If the saturation flux is too large，the nonlinear effect will be serious. Therefore，it is appropriate to select a SESAM with the saturation flux less than 50 μJ/cm² as a mode-locked device. Generally，a SESAM with the relatively small saturation flux is popular to obtain a mode-locked laser pulse with the even narrower pulse width.

Using theoretical simulation results analyzed by an approach of control variation，we finally determined all suitable parameters of an Yb-doped fiber laser. Next，we undertook the experimental verification using a system as shown in Fig. 14. A FBG（Ruike Huatai Electronic Tech. Corp. Ltd.）with the reflectance of 80%，reflection bandwidth of 0.31 nm，and center wavelength of 1.06 μm，was selected as the cavity output coupler. A gain fiber（Yb1200-4-125，n-LIGHT Inc.）with the length of 0.13 m and a SESAM（SAM-1064-28-15ps-x，German BATOP Ltd.）were utilized as the laser medium and mode-locked device，respectively. The modulation depth，saturation flux，and reflection bandwidth are 18%，30 μJ/cm²，and 70 nm，respectively. Such a SESAM is especially suitable for constructing a multi-band mode-locked laser. The transmission fiber（HI1060，Corning Inc.）is employed as tail fibers and a transmission jumper. Note that the length of a tail fiber was strictly controlled as short as possible.

When the power of the 976 nm pump LD was gradually increased from 0 to 24 mW，the unstable pulse lasing can be observed through an 8 GHz oscilloscope. When the pump power is increased to 27 mW，a pulse cluster is generated. When the pump power is increased to 40 mW，a stable mode-locked pulse output can be achieved with the repetition rate of 37 MHz. The output average power was measured as 0.45 mW by a power meter（Newport1830-R）. The spectral width of the mode-locked pulse was measured as 0.22 nm@3 dB at the center wavelength of 1.06 μm by using a spectrometer（AQ6373-10-H，YOKOGAWA Inc.）. The spectral width is much narrower than the theoretical results because the reflection bandwidth of the selected FBG is as narrow as 0.31 nm. The measured pulse sequence and spectrum are shown in Figs. 15（a）and（b）for stable mode locking，respectively. It can be seen that the spectral edge of the mode-locked pulse is quite smooth，and the overall shape is very close to the Gaussian distribution. The input-output power characteristics are plotted for our mode-locked fiber laser in Fig. 16.

Next，the pump power was gradually increased to 65 mW，and the output power is increased to 2 mW. After working for 5 hours，we observed that the mode-locked pulse maintained stably，and the output power has no obvious fluctuation. When the pump power was increased to 66 mW，the regular single pulse output changed to an irregular double pulse one for our mode-locked laser，and the time interval between two mode-locked pulses is about 0.027 μs，as shown in Fig. 17. The reason of presence of double pulses is thought to derive from the high-order harmonic mode-locking. Note that the spectral characteristics of the output pulse had no obvious variation at that time.

The spectra of the mode-locked pulses are also compared for different pump power as shown in Fig. 18. It is found that the 3 dB spectral width is slightly broadened from 0.22 to 0.32 nm with the increase of the pump power due to the nonlinear effect in the cavity. However，all the spectral curves remain the good Gaussian-shape.

Because the output power of the seed source is low and the peak power is not high enough，the pulse width cannot be detected by the autocorrelation instrument. In our study，the pre-amplifier module was added to enlarge the outputted power on purpose，and the pulse width can be therefore measured. The pump power of the seed and pre-amplifier modules are 60 and 100 mW，respectively. The output pulse width was measured as 12.51 ps by use of an autocorrelator（A·P·E Ltd.）as shown in Fig. 19. The pulse train after the pre-amplification was still stable，and the overall shape of the spectrum was confirmed to be less influenced by nonlinear effects such as amplified spontaneous emission（ASE）and self-phase modulation（SPM）. Considering the positive dispersion and nonlinear effects introduced by the pre-amplifier stage，the pulse width of the seed light should be less than 12.51 ps.

The mode field radius of a single-mode fiber，the length of a gain fiber，the reflectivity of a FBG，and the main parameters of a SESAM in the linear cavity are theoretically analyzed by the numerical simulation for a mode-locked fiber laser. A subtraction optimization of the laser structure has been realized based on the theoretical results，and a mode-locked fiber laser with the non-polarization maintaining linear cavity has been built for experimental verification. Finally，we obtained a satisfactory mode-locked laser source with the center wavelength of 1.06 μm，the pulse width of less than 12.51 ps，the 3 dB spectral width of 0.32 nm，the repetition rate of 37 MHz，and the output power of 2 mW，respectively. It is believed that such a mode-locked fiber laser might be valuable as a seed source for the construction of an industrial picosecond pulse laser system in the near future.