Patent Publication Number: US-2023163553-A1

Title: Fiber laser system

Description:
FIELD 
     The improvements generally relate to the field of fiber laser systems, and more particularly relate to optical pulse generation. 
     BACKGROUND 
     A fiber laser system generally has a laser-active optical fiber, a pump laser to pump the laser-active optical fiber, and a laser cavity supporting laser-modes when the laser-active optical fiber is pumped by the pump laser. When the laser-modes are locked to one another, the laser-modes are forced to constructively interfere with one another in a periodic manner, each time generating an optical pulse. Although existing fiber laser systems generating optical pulses are satisfactory to a certain degree, there always remains room for improvement, especially in limiting the number of free-space optical components, increasing the energy per optical pulse and/or generating shorter optical pulses. 
     SUMMARY 
     In an aspect, there is described a fiber laser system generally having a pump laser which generates a pump laser beam. The fiber laser system has a length of optical fiber which is optically coupled to the pump laser. The length of optical fiber has a laser cavity with an optical gain region, and two fiber Bragg gratings sandwiching the optical gain region. The two fiber Bragg gratings have reflectivity profiles which are detuned from one another. Accordingly, upon pumping of the optical gain region with the pump laser beam, and also upon mode locking of the laser cavity, only optical pulses that are amplified enough to experience spectral broadening as they propagate through the optical gain region are reflected back and forth within the laser cavity, between the two fiber Bragg gratings. For instance, noise that would be reflected by the reflectivity profile of the first fiber Bragg grating but not amplified enough to exhibit spectral broadening would not be reflected by the reflectivity profile of the second fiber Bragg grating, and vice versa. The fiber laser system has an output optically coupled to the laser cavity for outputting at least a portion of the optical pulses reflected within the laser cavity. 
     It was found that as conventional uniform fiber Bragg gratings have relatively narrow reflectivity profile bandwidths and relatively modest reflectivities, only a portion of each optical pulse can be effectively reflected within the laser cavity. Accordingly, it was found that by providing the first fiber Bragg grating with a first refractive index profile having a full width at half maximum bandwidth of at least 0.2 nm and a Gaussian-like apodization, optical pulses of a greater pulse energy and/or of a greater compressibility quality can be obtained. For instance, the Gaussian-like apodization can favor stability at greater pulse energies as this apodization can limit the presence of any side-lobes in the resulting reflectivity profile in turn increasing the desirability of the resulting optical pulses. In embodiments where the first refractive index profile has a varying grating period (i.e., a chirp), the first reflectivity profile can see its maximal reflectivity value increase, in addition to a broadening of its bandwidth, which can reflect a greater portion of the optical pulses present within the laser cavity. 
     In accordance with a first aspect of the present disclosure, there is provided a fiber laser system comprising: a pump laser generating a pump laser beam; and a length of optical fiber optically coupled to the pump laser, the length of optical fiber having: a laser cavity having a cavity path, a first fiber Bragg grating having a first reflectivity profile, a second filter having a second filter profile, and an optical gain region between the first fiber Bragg grating and the second filter along the cavity path, the first reflectivity profile being spectrally detuned from the second filter profile, the first fiber Bragg grating having a first refractive index profile comprising a full width at half maximum bandwidth of at least 0.2 nm and a Gaussian-like apodization, wherein, upon pumping of the optical gain region with the pump laser beam and mode locking of the laser cavity, optical pulses are circulated along the cavity path; and an output optically coupled to the laser cavity and outputting at least a portion of the optical pulses. 
     Further in accordance with the first aspect of the present disclosure, the first reflectivity profile of the first fiber Bragg grating can for example have a maximal reflectivity value of at least 40% and the full width at half maximum bandwidth is of at least 0.5 nm. 
     Still further in accordance with the first aspect of the present disclosure, the maximal reflectivity value of the first reflectivity profile can for example be at least 50%. 
     Still further in accordance with the first aspect of the present disclosure, the full width at half maximum bandwidth of the first reflectivity profile can for example be between about 4 nm and about 5 nm. 
     Still further in accordance with the first aspect of the present disclosure, the first refractive index profile can for example have a varying grating period, i.e., be chirped. 
     Still further in accordance with the first aspect of the present disclosure, the varying grating period of a respective one of the first and second refractive index profile can for example vary in a linear manner, thereby offering a linear group delay dispersion across the respective one of the first reflectivity profile and the second filter profile. 
     Still further in accordance with the first aspect of the present disclosure, the linear group delay dispersion can for example be steeper than ±0.5 ps 2 . 
     Still further in accordance with the first aspect of the present disclosure, the linear group delay dispersion of the varying grating period can for example be at least twice as steep as a linear group delay dispersion offered by the optical gain region. 
     Still further in accordance with the first aspect of the present disclosure, the second filter profile can for example have a maximal reflectivity or transmissivity value being smaller than a maximal reflectivity value of the first reflectivity profile, the output being optically coupled to the second filter. 
     Still further in accordance with the first aspect of the present disclosure, the second filter can for example be a second fiber Bragg grating, the second filter profile being a second reflectivity profile spectrally detuned from the first reflectivity profile. 
     Still further in accordance with the first aspect of the present disclosure, the first and second fiber Bragg gratings can for example sandwich at least a portion of the optical gain region along the cavity path, the cavity path thereby being a linear path along which the optical pulses are reflected in back-and-forth between the first and second fiber Bragg gratings. 
     Still further in accordance with the first aspect of the present disclosure, the second fiber Bragg grating can for example have a second refractive index profile having a full width at half maximum bandwidth of at least 0.2 nm and a Gaussian-like apodization. 
     Still further in accordance with the first aspect of the present disclosure, the second reflectivity profile of the second fiber Bragg grating can for example have a maximal reflectivity value of at least 40% and the full width at half maximum bandwidth is of at least 0.5 nm. 
     Still further in accordance with the first aspect of the present disclosure, the maximal reflectivity value of the second reflectivity profile can for example be at least 50%. 
     Still further in accordance with the first aspect of the present disclosure, the full width at half maximum bandwidth of the second reflectivity profile can for example be between about 4 nm and about 5 nm. 
     Still further in accordance with the first aspect of the present disclosure, the second refractive index profile can for example have a varying grating period. 
     Still further in accordance with the first aspect of the present disclosure, the fiber laser system can for example further comprise a mode locking device coupled to the length of optical fiber and contributing to said mode locking of said laser cavity. 
     Still further in accordance with the first aspect of the present disclosure, the mode locking device can for example have a stretching element longitudinally stretching at least one of the first fiber Bragg grating and the second filter, thereby modifying a spectral detuning between the first reflectivity profile and the second filter profile. 
     Still further in accordance with the first aspect of the present disclosure, the mode locking device can for example have an optical coupler optically coupled to the length of optical fiber and leading to a mode locking arm. 
     Still further in accordance with the first aspect of the present disclosure, the optical gain region can for example be a laser-active erbium-doped region, the pump laser beam having a wavelength of about 980 nm, the first reflectivity profile having a central wavelength of about 1550 nm, the second filter profile having a central wavelength of about 1565 nm. 
     Still further in accordance with the first aspect of the present disclosure, the pump laser can for example be a first pump laser, the fiber laser system further comprising a second pump laser optically coupled to the length of optical fiber and propagating a second pump laser beam along said optical gain region during said pumping. 
     Still further in accordance with the first aspect of the present disclosure, the output can for example be a main output optically coupled to the second filter, the fiber laser system comprising an auxiliary output optically coupled to the first fiber Bragg grating. 
     Still further in accordance with the first aspect of the present disclosure, the outputted optical pulses can for example have a similariton-like profile having a linearly varying instantaneous frequency and a pulse duration below 100 fs after compression. 
     Still further in accordance with the first aspect of the present disclosure, the optical pulses can for example have a pulse energy of at least 10 nJ. 
     In accordance with a second aspect of the present disclosure, there is provided a laser system comprising: a pump laser generating a pump laser beam; a laser cavity optically coupled to the pump laser, the laser cavity having a cavity path, a first filter having a first filter profile of a Gaussian-like shape, a second filter having a second filter profile, and an optical gain region between the first and second filters along the cavity path, the first and second filter profiles being spectrally detuned from one another, the first filter being dispersive thereby imparting a dispersive profile across at least at portion of the first filter profile, wherein, upon pumping of the optical gain region with the pump laser beam and mode locking of the laser cavity, optical pulses are circulated along the cavity path; and an output optically coupled to the laser cavity and outputting at least a portion of the optical pulses. 
     Further in accordance with the second aspect of the present disclosure, the first filter can for example be a fiber Bragg grating having an refractive index profile comprising a full width at half maximum bandwidth of at least 0.2 nm and a Gaussian-like apodization. 
     Still further in accordance with the second aspect of the present disclosure, the refractive index profile can for example have a varying grating period. 
     Still further in accordance with the second aspect of the present disclosure, at least a portion of the laser cavity can for example be fibered. 
     Many further features and combinations thereof concerning the present improvements will appear to those skilled in the art following a reading of the instant disclosure. 
    
    
     
       DESCRIPTION OF THE FIGURES 
       In the figures, 
         FIG.  1    is a schematic view of an example of a fiber laser system, with a pump laser, an optical gain region, and first and second fiber Bragg gratings sandwiching the optical gain region, in accordance with one or more embodiments; 
         FIG.  1 A  is a graph showing an example of a first reflectivity profile of the first fiber Bragg grating of  FIG.  1   , also showing a group delay varying across at least a portion of the first reflectivity profile, in accordance with one or more embodiments; 
         FIG.  1 B  is a graph showing an example of a second reflectivity profile of the second fiber Bragg grating of  FIG.  1   , also showing a constant group delay, in accordance with one or more embodiments; 
         FIG.  10    is a graph showing another example of a second reflectivity profile of the second fiber Bragg grating of  FIG.  1   , also showing a group delay varying across at least a portion of the second reflectivity profile, in accordance with one or more embodiments; 
         FIG.  2    is a graph showing an exemplary refractive index profile of the first fiber Bragg grating of  FIG.  1   , showing a chirp and a Gaussian-like apodization, in accordance with one or more embodiments; 
         FIG.  3    is a schematic view of the fiber laser system of  FIG.  1   , shown with a mode locking device comprising a stretching element stretching the second fiber Bragg grating, in accordance with one or more embodiments; 
         FIG.  3 A  is a graph showing reflectivity profiles of the first and second fiber Bragg gratings of the fiber laser system of  FIG.  1   , in accordance with one or more embodiments; 
         FIG.  4    is a schematic view of another example of a fiber laser system, with a pump laser, an optical gain region, first and second fiber Bragg gratings sandwiching the optical gain region, a mode locking arm and a pump reflector, in accordance with one or more embodiments; 
         FIG.  4 A  is a schematic view of a first example of the mode locking arm of  FIG.  4   , in accordance with one or more embodiments; 
         FIG.  4 B  is a schematic view of a second example of the mode locking arm of  FIG.  4   , in accordance with one or more embodiments; 
         FIG.  5    is a schematic view of another example of a fiber laser system, with two pump lasers, an optical gain region, first and second fiber Bragg gratings sandwiching the optical gain region, and a mode locking arm, in accordance with one or more embodiments; 
         FIG.  6    is a schematic view of an example of a fiber laser system with a ring-type laser cavity incorporating first and second fiber Bragg gratings, in accordance with one or more embodiments; 
         FIG.  7    is a schematic view of another example of a fiber laser system with a ring-type laser cavity incorporating a first fiber Bragg grating and a second dielectric transmissive filter, in accordance with one or more embodiments; 
         FIG.  8    is a schematic view of another example of a fiber laser system, with a laser cavity having a fiber-active laser region being sandwiched between first and second chirped fiber Bragg gratings, with first and second mode locking devices, in accordance with an embodiment; 
         FIG.  8 A  is a graph showing a first reflectivity profile of the first chirped fiber Bragg grating of  FIG.  8   , with a full width at half maximum of 4.7 nm and a maximal reflectivity value of 65%, and showing a group delay dispersion of −0.87 ps 2 ; 
         FIG.  8 B  is a graph showing a second reflectivity profile of the second chirped fiber Bragg grating of  FIG.  8   , with a full width at half maximum of 4.5 nm and a maximal reflectivity value of 41%, and showing a group delay dispersion of −0.82 ps 2 ; 
         FIG.  9 A  is a graph showing simulated and experimental spectra of optical pulses generated by the fiber laser system of  FIG.  8   ; 
         FIG.  9 B  is a graph showing simulated and experimental autocorrelation traces of optical pulses generated by the fiber laser system of  FIG.  8   ; 
         FIG.  9 C  is a graph showing simulated and measured pulse envelope after compression for optical pulses generated by the fiber laser system of  FIG.  8   ; 
         FIG.  9 D  is a graph showing a radio frequency spectrum compared to noise for optical pulses generated by the fiber laser system of  FIG.  8   ; 
         FIG.  10 A  is a graph showing simulations for peak power and pulse energy for optical pulses after compensation of the second-order dispersion as a function of the first chirped fiber Bragg grating for a second chirped fiber Bragg grating having a maximal reflectivity value of 41%; 
         FIG.  10 B  is a graph showing simulations for peak power and pulse energy for optical pulses after compensation of the second-order dispersion as a function of the first chirped fiber Bragg grating for a second chirped fiber Bragg grating having a maximal reflectivity value of 15%; 
         FIG.  11 A  is a graph showing an example evolution of a pulse parabolic misfit parameter M 2 =∫(I−I fit ) 2 dt/∫I 2 dt as a function of the length of the fiber-active doped region of the fiber laser system of  FIG.  8   ; 
         FIG.  11 B  is a graph showing pulse and gain spectra of optical pulses at a first longitudinal position along the length of the fiber-active doped region of the fiber laser system of  FIG.  8   ; 
         FIG.  11 C  is a graph showing pulse and gain spectra of optical pulses at a second longitudinal position spaced-apart from said first longitudinal position; 
         FIG.  12 A  is a graph showing another example of a refractive index profile of a fiber Bragg grating operating at 1552 nm in an erbium-based laser cavity, in accordance with one or more embodiments; 
         FIG.  12 B  is a graph showing an example reflectivity profile of the fiber Bragg grating of  FIG.  12 A , in accordance with one or more embodiments; 
         FIG.  13 A  is a graph showing another example of a refractive index profile of a fiber Bragg grating operating at 1050 nm in an ytterbium-based laser cavity, in accordance with one or more embodiments; and 
         FIG.  13 B  is a graph showing an example reflectivity profile of the fiber Bragg grating of  FIG.  13 A , in accordance with one or more embodiments. 
     
    
    
     DETAILED DESCRIPTION 
       FIG.  1    shows an example of a fiber laser system  100 . As depicted, the fiber laser system  100  has a pump laser  102  generating a pump laser beam  104 , and a length  106  of optical fiber which is optically coupled to the pump laser  102 . 
     The length  106  of optical fiber has a laser cavity  108  having a cavity path  109 , an optical gain region  110 , a first fiber Bragg grating  112  along the cavity path  109 , a second filter  111  spaced-apart from the first fiber Bragg grating  112  along the cavity path  109 , in which the optical gain region  110  lies between the first fiber Bragg grating  112  and the second filter  111 . The length  106  of optical fiber also has an output  116  which is optically coupled to the laser cavity  108  for outputting optical pulses  118  generated within the laser cavity  108 . 
     In this specific embodiment, the second filter  111  is provided in the form of a second fiber Bragg grating  114 . As shown, the first and second fiber Bragg gratings  112  and  114  sandwich the optical gain medium  110 , in which case the cavity path  109  is a linear path along which the optical pulses  118  are reflected in back-and-forth between the first and second fiber Bragg gratings  112  and  114 . 
     Each of the first and second fiber Bragg gratings  112  and  114  has a respective refractive index profile inscribed along respective portions  112   a  and  114   a  of the length  106  of optical fiber, thereby imparting a corresponding reflectivity profile to the first and second fiber Bragg gratings  112  and  114 . 
     In this specific example, the first fiber Bragg grating  112  has a first reflectivity profile which is detuned from a second reflectivity profile of the second fiber Bragg grating.  FIG.  1 A  shows an example of a first reflectivity profile  122  of the first fiber Bragg grating  112 , which is in this case centered on a first Bragg wavelength λ 1 .  FIGS.  1 B and  1 C  show examples of a second reflectivity profile  124  of the second fiber Bragg grating, which are in this case centered on a second Bragg wavelength λ 2  being spectrally spaced-apart from the first Bragg wavelength λ 1  of the first reflectivity profile  122 . 
     As can be appreciated, the spectral contents of an optical pulse reflected by the first reflectivity profile  122  of the first fiber Bragg grating  112  tend to be spectrally broadened (see arrow A) during its propagation in the laser cavity  108  to encompass the second Bragg wavelength λ 2  of the second reflectivity profile  124  of the second fiber Bragg grating  114 , and vice versa, thereby allowing optical pulses to be reflected in back-and-forth between the first and second fiber Bragg gratings  112  and  114  when the optical gain region  110  is pumped with the pump laser beam  104  and when the laser cavity  108  is mode locked. As such, only optical pulses that are amplified enough to experience spectral broadening as they propagate through the optical gain region  110  can oscillate within the laser cavity  108 . Such architecture, based on the presence of two offset filters in a nonlinear laser cavity, can be referred to as a Mamyshev oscillator. 
     In this example, the first fiber Bragg grating  112  has a first refractive index profile  126  comprising a varying grating period  128  (i.e., a chirp) and a Gaussian-like apodization  130 , an example of which is shown in  FIG.  2   . As can be appreciated, the grating period  128  of the first refractive index profile varies along the portion of the length of optical fiber on which it is inscribed. The varying grating period creates a chirp, thereby enhancing a maximal reflectivity value of the first reflectivity profile and broadening its bandwidth. In this embodiment, the first fiber Bragg grating is thus a chirped fiber Bragg grating or “CFBG.” The Gaussian-like apodization  130  shows that the refractive index n varies according to a Gaussian-like profile. The Gaussian-like profile favors stable optical pulses at greater pulse energy. The term apodization refers to the grading of the refractive index n which approaches zero at both ends  130   a  and  130   b  of the first refractive index profile  126 . Apodized fiber Bragg gratings can be convenient in terms of side-lobe suppression while maintaining satisfactory reflectivity. The Gaussian-like apodization  130  can be described as encompassing any apodization which can achieve a smooth reflectivity profile within the main lobe such that the rate of variations in absolute is below about 1000%/nm and/or side-lobes reflectivity are below about −10 dB. In some embodiments, the Gaussian-like apodization  130  can have a shape that is slightly asymmetric which can yield a symmetric Gaussian reflectivity profile, for instance. It is noted that the first refractive index profile  126  shown in  FIG.  2    is exemplary only, as the illustrated varying grating period is not a real value nor to scale. The varying grating period  128 , or equivalently the chirp, is only optional in the first refractive index profile  126 . Accordingly, the first refractive index profile  126  can have a non-varying or uniform grating period in some other embodiments. 
     The construction of the first fiber Bragg grating is such that the first refractive index profile  126  has a full width at half maximum bandwidth of at least 0.2 nm, which can provide greater pulse energy and/or greater compressibility quality. In some embodiments, the full width at half maximum bandwidth of the first refractive index profile  126  can advantageously be higher than 0.2 nm. Referring back to  FIG.  1 A , thanks to the first varying grating period and to the Gaussian-like apodization of the first refractive index profile, the first reflectivity profile  122  of the first fiber Bragg grating  112  can have a maximal reflectivity value R 1  of at least 40% and a full width at half maximum bandwidth Δλ 1  of at least 0.5 nm. In some embodiments, the maximal reflectivity value R 1  of the first reflectivity profile  122  is at least 50%, preferably at least 55%, and most preferably at least 60%. In some embodiments, the full width at half maximum bandwidth Δλ 1  of the first reflectivity profile  122  is at least about 0.2 nm, preferably at least about 2 nm or 3 nm and most preferably between about 4 nm and about 5 nm. 
     It is encompassed that the optical pulses  118  that are outputted at the output  116  can have a pulse energy of at least 10 nJ, preferably at least 15 nJ and most preferably at least 20 nJ, thanks to the maximal reflectivity value R 1  and to the full width at half maximum Δλ 1  of the first reflectivity profile  122  of the first fiber Bragg grating  122 . 
     In some embodiments, the varying grating period of the first fiber Bragg grating  112  can vary in a linear manner across at least a portion of the first reflectivity profile  122 , as in Fabry-Perot laser cavities. In these embodiments, the first fiber Bragg grating  112  can offer a linear group delay dispersion across at least a portion of the first reflectivity profile  122 , such as shown by the dashed line of  FIG.  1 A . It is envisaged that the linear group delay dispersion can be steeper than ±0.5 ps 2 , example of which will be described below. In some embodiments, the linear group delay dispersion of the varying grating period of the first fiber Bragg grating  112  is at least twice as steep as a linear group delay dispersion offered by the optical gain region  110 . In some other embodiments, the varying grating period of the first fiber Bragg grating  112  can vary in a nonlinear manner as well. Regardless of how the grating period of the first fiber Bragg grating  112  varies, the first fiber Bragg grating  112  can be used to compensate dispersion occurring elsewhere within the laser cavity  108 . For instance, the optical gain region  110  may have a normal dispersion, i.e., the optical group velocity increases with wavelength, which will cause an optical pulse propagating thereacross to see its longer wavelengths travel faster than its shorter wavelengths. Anomalous dispersion is just the opposite, meaning that the group velocity of the guided optical mode decreases with a gradually increasing wavelength. The cavity dispersion at each round-trip does not need to be strictly compensated. However, the group velocity dispersion of the first fiber Bragg grating  112  can be used to compensate the fiber dispersion and manipulate the filtered pulse phase, intensity, and spectrum profile. Thus, it is possible to optimize the following nonlinear amplification which can, for instance, improve the output pulse compressibility. The first fiber Bragg grating  112  can be used to impart this opposite-sign dispersion to an optical pulse. As such, the linear group delay dispersion can be anomalous in some embodiments, whereas the linear group delay dispersion can be normal in some other embodiments. When the opposite-sign is chosen, a higher absolute value of group delay dispersion can be used for the first fiber Bragg grating  112 , meaning lower chirp for the index profile of the first fiber Bragg grating  112 , and thus further improve the maximal reflectivity value R 1  of the first reflectivity profile. 
     In some embodiments, the second fiber Bragg grating  114  has a second refractive index profile comprising a constant grating period (i.e., no chirp), thereby imparting a reflectivity profile such as the one shown at  124  in  FIG.  1 B . In such embodiment, the second reflectivity profile  124  of the second fiber Bragg grating  114  can have a reduced reflectivity value R 2  and a reduced full width at half maximum bandwidth Δλ 2 . In some embodiments, the maximal reflectivity value R 2  of the second reflectivity profile  124  is at least 10%, preferably at least 20%, and most preferably at least 30%. In some embodiments, the full width at half maximum bandwidth Δλ 2  of the second reflectivity profile  124  is at least about 2 nm, preferably at least about 3 nm and most preferably between about 4 nm and about 5 nm. In embodiment where the second fiber Bragg grating  114  is not chirped, the second fiber Bragg grating  114  can offer a null group delay dispersion across at least a portion of the second reflectivity profile  124 , such as shown by the dashed line of  FIG.  1 B . In any case, the full width at half maximum bandwidths of the first and second reflectivity profiles are similar to one another in some embodiments. 
     Although not mandatory, the second fiber Bragg grating  114  can be chirped in a similar way than the first fiber Bragg grating  112 . For instance, the grating period of the second refractive index profile can also vary along the portion of the length of optical fiber on which it is inscribed. The varying grating period creates a chirp, thereby enhancing a maximal reflectivity value R 2  of the second reflectivity profile  124  and broadening its bandwidth Δλ 2 , such as shown in the second reflectivity profile shown at  124  in  FIG.  10   . The Gaussian-like apodization of the second refractive index profile can favor stable optical pulses or greater pulse energy. In such embodiments, the second refractive index profile can have similarities with the first refractive index profile. The second reflectivity profile  124  can also have a full width at half maximum bandwidth of at least 0.2 nm in some embodiments. 
     Still referring to  FIG.  10   , the second reflectivity profile  124  of the second fiber Bragg grating  114  can have a maximal reflectivity value R 2  of at least 40% and a full width at half maximum bandwidth Δλ 2  of at least 0.5 nm due to the varying grating period and Gaussian-like apodization of the second optical fiber refractive index. In some embodiments, the maximal reflectivity value R 2  of the second reflectivity profile  124  is at least 50%, preferably at least 55%, and most preferably at least 60%. In some embodiments, the full width at half maximum bandwidth Δλ 2  of the second reflectivity profile  124  is at least about 0.2 nm, preferably at least about 2 nm or 3 nm, and most preferably between about 4 nm and about 5 nm. 
     In some embodiments, the varying grating period of the second fiber Bragg grating  114  can vary in a linear manner across at least a portion of the second reflectivity profile  124 . In these embodiments, the second fiber Bragg grating  114  can offer a linear group delay dispersion across at least a portion of the second reflectivity profile  114 , such as shown by the dashed line of  FIG.  1 C . It is envisaged that the linear group delay dispersion can be steeper than ±0.5 ps 2 . In some embodiments, the linear group delay dispersion of the varying grating period of the second fiber Bragg grating  114  is at least twice as steep as a linear group delay dispersion offered by the optical gain region  110 . In some other embodiments, the varying grating period of the second fiber Bragg grating  114  can vary in a nonlinear manner. The linear group delay dispersion can be negative in some embodiments, whereas the linear group delay dispersion can be positive in some other embodiments. 
     In some embodiments, the second filter  111  may not be a fiber Bragg grating. Regardless, the second filter  111  imparts a filter profile which is spectrally detuned from the first reflectivity profile  122  of the first fiber Bragg grating  112 , thereby forming an oscillator of the Mamyshev type. Accordingly, the second filter  111  can be reflective or transmissive in which case the corresponding second filter profile can be a reflectivity profile or a transmissivity profile, respectively. Indeed, in these embodiments, the second filter  111  can be a dielectric filter, a tunable filter, a loop incorporating one or more filter units and one or more optical circulators, or any combination thereof. In some embodiments, however, providing the second filter  111  in the form of a fiber Bragg grating may be convenient as it may allege the complexity of the corresponding laser cavity. 
     As the example illustrated in  FIG.  1   , the length  106  of optical fiber comprises a first segment  106   a  of optical fiber having the first fiber Bragg grating  112 , a second segment  106   b  having the optical gain region  110 , a third segment  106   c  having the second fiber Bragg grating  114 , and a fourth segment  106   d  having the output  116 . In this specific example, the first, second, third and fourth segments are fusion spliced or otherwise optically connected to one another. For instance, optical connectors can be used to optically connect the segments to one another. Depending on the embodiment, the length  106  of optical fiber can have fewer than or more than 4 segments. For instance, the first and second fiber Bragg gratings  112  and  114  can be inscribed directly within the optical gain region  110  in some embodiments. It is noted that the length  106  of optical fiber can have optical fiber segments other than those for the first and second fiber Bragg gratings  112  and  114  and for the optical gain region  110 . Indeed, it is envisaged that the length  106  of optical fiber can include any suitable number of optical fiber segments comprising either passive or active optical fiber. In some embodiments, additional passive optical fiber segments are used within the laser cavity  108  to control parameters such as the dispersion, the nonlinearity and the repetition rate of the resulting fiber laser system. 
     Also shown in this embodiment, the pump laser  102  is provided in the form of a fibered laser diode  132  which fibered output  134  is optically connected to the first segment  106   a  of the length  106  of optical fiber. More specifically, the fibered output  134  of the fibered laser diode  132  is fusion spliced or otherwise optically connected to the first end  106   a  of the length  106  of optical fiber in this specific embodiment. In some other embodiments, the pump laser  102  can be provided in any suitable type of laser including, but not limited to, a fiber laser emitting a pump laser beam, and the like. 
     As can be appreciated, in this example, the second reflectivity profile  114  has a maximal reflectivity value R 2  which is smaller than a maximal reflectivity value R 1  of the first reflectivity profile  112 , in this case the output  116  is optically coupled to the second fiber Bragg grating  114  which transmits a greater portion of the optical pulses  118  oscillating within the laser cavity  108  than the first fiber Bragg grating  112  would. In some embodiments, the outputted optical pulses  118  have a similariton-like profile  136  having a linearly varying instantaneous frequency thereby allowing a pulse duration below 100 fs or below after compression. 
     The optical gain region  110  can be any type of optical fiber being laser-active via doping with one or more rare-earth ions such as erbium ions (Er 3+ ), ytterbium ions (Yb 3+ ), thulium ions (Tm 3+ ), holmium ions (Ho 3+ ), dysprosium ions (Dy 3+ ), praseodymium ions (Pr 3+ ), neodymium ions (Nd 3+ ) or any combination thereof. The concentration of rare-earth ions can vary from a laser-active doped region to another. The laser-active doped region can have a conventional silica-based matrix embedding the rare-earth ions. Otherwise, the matrix of the optical fiber can be a low phonon energy glass such as fluoride-, chalcogenide-, chalcohalide- or telluride-based glass to name a few examples. For instance, in some embodiments, the low phonon energy glass is a zirconium fluoride glass having a composition including ZrF 4  such as ZBLAN (ZrF 4 /HfF 4 , BaF 2 , LaF 3 , NaF, and AlF 3 ). In some other embodiments, the low phonon energy glass is an indium fluoride glass having a composition including InF 3 . In alternate embodiments, the low phonon energy glass is an aluminum fluoride glass having a composition including AlF 3 . In further embodiments, the low phonon energy glass is a chalcogenide glass having a composition including As 2 S 3 , As 2 Se 3 , AsTe, AsSSe, AsSTe, GaLaS, GeAsS, GeAsSe or the like. Photonic crystal fibers, large mode area (LMA) fiber, and other type of specialty optical fiber may be used in the fiber laser system. Moreover, it is noted that the optical gain region  110  can provide optical gain via nonlinear effects such as stimulated Raman scattering or any other suitable nonlinear effect or combination thereof. In embodiments where the optical gain region  110  relies on such nonlinear effects, the optical gain region  110  needs not to be doped with rare-earth ions, for instance. 
     For instance, in embodiments where the optical gain region  110  is erbium-doped with a silica-based matrix, the pump laser beam  104  can have a wavelength of about 980 nm. The length of the optical gain region  110  can be at least 5 m, preferably at least 8 m, and most preferably at least 10 m. For instance, in this specific embodiment, the optical gain region  110  is 10.8-m long. In these embodiments, the first reflectivity profile  122  can have a first Bragg wavelength λ 1  of about 1550 nm whereas the second reflectivity profile  124  can have a Bragg wavelength λ 2  of about 1565 nm, thereby leaving a spectral detuning of about 15 nm. In some embodiments, the spectral detuning can be less than about 15 nm whereas in some other embodiments, the spectral detuning can be more than about 15 nm. 
     In this specific embodiment, the first fiber Bragg grating  112  is chirped, with a first reflectivity profile having a maximal reflectivity R 1  of about 65% and a full width at half maximum Δλ 1  of about 4.7 nm. The second fiber Bragg grating  114  is chirped, too, with a second reflectivity profile having a maximal reflectivity R 2  of about 41% and a full width at half maximum Δλ 2  of about 4.5 nm. The length  106  of optical fiber is entirely made of polarization-maintaining (PM) fiber in this embodiment, which can contribute to the stability of the optical pulse generation. However, in some other embodiments, only a portion of the length  106  of optical fiber can be PM. 
     As will be understood from the example below, the fiber laser system described herein can have one or more mode locking devices, one or more pump lasers, one or more outputs, one or more optical couplers, one or more polarization isolators, pump reflectors and the like. 
     For instance,  FIG.  3    shows the fiber laser system  100  with an example mode locking device  140  which is coupled to the length  106  of optical fiber and which contributes to the mode locking of the laser cavity  108 . As shown in this example, the mode locking device  140  has two spaced-apart stretching elements  142  longitudinally stretching the first fiber Bragg grating  112 , thereby modifying a spectral detuning between the first and second reflectivity profiles  122  and  124 , which can facilitate the mode locking.  FIG.  3 A  shows how the first and second reflectivity profiles  122  and  124  can be brought together or pushed away from one another using the stretching elements  142 . In some circumstances, to mode lock the laser cavity  108 , the stretching elements  142  are operated so that adjacent tails  146  of the first and second reflectivity profiles  122  and  124  overlap with one another by a given amount. Alternatively or additionally, a similar mode locking device can be used to stretch the second fiber Bragg grating  114 . In any case, stretching a fiber Bragg grating has a tendency to redshift the Bragg wavelength of the corresponding reflectivity profile. Accordingly, if the first and second reflectivity profiles are to be brought towards one another, the stretching elements should be preferably installed on the fiber Bragg grating having the lower Bragg wavelength. In this case, the first fiber Bragg grating  112  has a first Bragg wavelength λ 1  which is lower than a second wavelength λ 2  of the second fiber Bragg grating  114 , and accordingly the stretching elements  142  stretch the first fiber Bragg grating  112 . In some other embodiments, the second wavelength λ 2  of the second fiber Bragg grating  114  is lower than the first Bragg wavelength λ 1  of the first fiber Bragg grating  112  and thereby the stretching elements  142  are installed on the second fiber Bragg grating  114  instead of on the first fiber Bragg grating  112 . In some embodiments, the fiber laser system  100  has a tilted fiber Bragg grating  143  in the cavity path within the laser cavity  108 . The tilted fiber Bragg grating  143  can provide a polarizing effect and/or additional filtering within the laser cavity  108 , if necessary. 
       FIG.  4    shows another example of a fiber laser system  200 . As shown, the fiber laser system  200  has a pump  202 , a length  206  of optical fiber which is optically coupled to the pump  202 . The length  206  of optical fiber has a laser cavity  208  with an optical gain region  210  and first and second Bragg gratings  212  and  214  such as the ones described above, and an output  216  optically coupled to the length  206  of optical fiber. As shown in this specific example, the fiber laser system  200  has a mode locking device  240  with an optical coupler  246  tapped on the length  206  of optical fiber, upstream from the laser cavity  208 , and leading to a mode locking arm.  FIGS.  4 A and  4 B  show two different examples of such a mode locking devices  240 . As can be appreciated, the mode locking devices  240  of  FIGS.  4 A and  4 B  help mode locking the laser cavity  208 . In these embodiments, the fiber laser system  200  can have an auxiliary output  248  within the mode locking device. 
     The mode locking arm  240 ′ of  FIG.  4 A  has a first half wave plate  270 , a first polarization beam splitter  272 , a second half wave plate  274 , a second polarization beam splitter  276 , a flip mirror  278  movable between an in-path position and an out-of-path position, and a static mirror  280 . These components are arranged in series with one another. When the flip mirror  278  is moved in the in-path position, an optical signal outputted from the auxiliary output  248  will be propagated across the first half wave plate  270 , the first polarization beam splitter  272 , the second half wave plate  274 , the second polarization beam splitter  276 , and then back into the laser cavity via the auxiliary output  248 , thereby providing mode locking feedback to the laser cavity. Once the flip mirror  278  is moved back to the out-of-path position, the laser cavity can transition toward a stable single pulse mode locked regime. 
       FIG.  4 B  shows another example of a mode locking arm  240 ″. As depicted, the mode locking arm  240 ″ has a flip mirror  278 , a lens  282 , and a saturable absorber mirror  284 . When the flip mirror  278  is moved in the in-path position, an optical signal outputted from the auxiliary output  248  will be propagated across the lens  282  towards the saturable absorber mirror  284 , and then back into the laser cavity via the auxiliary output  248 , thereby providing mode locking feedback to the laser cavity. In a similar manner, once the flip mirror  278  is moved back to the out-of-path position, the laser cavity can transition toward a stable single pulse mode locked regime. 
     The mode locking devices  140  and  240  described above with reference to  FIGS.  3  and  4    are exemplary only. For instance, other mode locking devices can be used. In some embodiments, the mode locking device is a pump laser modulation device which modulates the pump laser  102  in accordance with predetermined modulation parameters, thereby providing mode locking feedback to the laser cavity  108  once steady state is achieved. Any other passive or active mode locking device can be used, as see fit by the skilled reader. The mode locking devices  140  and  240  described above with reference to  FIGS.  3  and  4    can also be omitted, as the fiber laser cavity may mode lock on its own depending on the circumstances. 
     Referring back to  FIG.  4   , the first and second fiber Bragg gratings  212  and  214  sandwich only a portion of the optical gain region  210 . For instance, the second fiber Bragg grating  214  is inscribed on a portion of the optical gain region  210  of the length  206  of optical fiber. Accordingly, the optical gain region  210  extends beyond the second fiber Bragg grating  214  towards the output  216  in this specific embodiment. As such, the length  206  of optical fiber can lack the optical losses generally associated with the optical connections between the laser cavity  208  and the output  216 . In addition, partial amplification of the optical pulses  218  can occur even outside the laser cavity  208  with this architecture. 
     Also shown in this embodiment, the pump laser beam  204  is propagating across the optical gain region  210  and towards the output  216 , and then reflected back towards the optical gain region  210  using a pump reflector  250  downstream from the second fiber Bragg grating  214 . As such, the pump reflector can reflect any remaining portion of the pump laser beam  204  back towards the optical gain region  210  for further pumping. The pump reflector  250  can be a fiber Bragg grating, for instance. The pump reflector  250  is only optional, as it can be omitted in some embodiments. 
     In the above-described embodiments, the fiber laser system  200  is made of single-clad fiber having a core surrounded by at least a cladding, in which case the first and second fiber Bragg gratings are inscribed within a core of the single-clad fiber. However, in some other embodiments, another exemplary fiber laser system can be made of multi-clad fiber having a core surrounded by an inner cladding which is in turn surrounded by at least an outer cladding. In these latter embodiments, as the first and second fibre Bragg gratings may be inscribed in the core of the multi-clad fiber, any pump reflector such as pump reflector  250  may be a fiber Bragg grating inscribed instead in the inner cladding of the multi-clad fiber. 
     In some embodiments, it may be envisaged to provide the laser cavity  208  with a fibered polarizer  260  between the first and second fiber Bragg gratings  212  and  214  to fix the polarization state of the optical pulses  218  as desired. In such embodiments, it may also be preferable to position the fibered polarizer  260  immediately downstream from the second fiber Bragg grating  214 . 
       FIG.  5    shows another example of a fiber laser system  300 . As shown, the fiber laser system  300  has a pump  302 , a length  306  of optical fiber optically coupled to the pump  302 . The length  306  of optical fiber has a laser cavity  308  with an optical gain region  310  and first and second Bragg gratings  312  and  314  such as the ones described above, and an output  316  optically coupled to the length  306  of optical fiber. In contrast with the fiber laser system  300  of  FIG.  4   , in this example, the pump laser  302  is downstream from the laser cavity  308 , and propagates the pump laser beam  304  away from the output  316  and towards the laser cavity  308 . In some embodiments, the fiber laser system  300  can have a second pump laser  302 ′ upstream from the laser cavity  308  and which is optically coupled to the length  306  of optical fiber via an optical coupler. In this embodiment, the second pump laser  302 ′ propagates a second pump laser beam  304 ′ along the optical gain region  310  for increased pumping results. In such embodiments, the fiber laser system  300  may be provided with one or more optical isolators within or outside the laser cavity  308  to protect the pump lasers  302  and  302 ′. This is particularly convenient when the pump laser beams  304  and  304 ′ have different wavelengths. 
     Although the fiber laser systems  100 ,  200  and  300  described above all have a linear cavity path along which the optical pulses are reflected in back-and-forth between the first and second fiber Bragg gratings, other embodiments of the fiber laser system described herein can have a ring-type cavity around which the optical pulses are circulated, examples of which are described with reference to  FIGS.  6  and  7   . 
       FIG.  6    shows an example of a fiber laser system  400  in accordance with a ring-type Mamyshev oscillator architecture. As depicted, the fiber laser system  400  has a first pump laser  402   a  and a length  406  of optical fiber which is optically coupled to the first pump laser  402   a  via a first wavelength division multiplexing (WDM) coupler  446   a.  The length  406  of optical fiber has a laser cavity  408  with at least a ring cavity path  409 , a first fiber Bragg grating  412  along the circular cavity path  409 , a second fiber Bragg grating  414  along the circular cavity path  409  spaced apart from the first fiber Bragg grating  412 , and a first optical gain region  410   a  between the first and second fiber Bragg gratings  412  and  414 . An output  416  is optically coupled to the ring-type laser cavity  408  for outputting optical pulses  418 . 
     As illustrated, optical circulators  486   a  and  486   b  are used to circulate optical pulses  418  reflected by the respective first and second fiber Bragg gratings  412  and  414  back on a different optical fiber segment, thereby allowing the optical pulses  418  to travel around the ring cavity path  409  in a counter-clock wise manner. Optical isolators  488  may be provided to block optical pulses propagating in the opposite direction. In this specific embodiment, a second optical gain region  410   b  is provided along the length  406  of optical fiber, between the first and second fiber Bragg gratings  412  and  414 . To pump the first and second optical gain regions  410   a  and  410   b,  two additional pump lasers  402   b  and  402   c  are optically coupled to the laser cavity  408  using respective WDM couplers  446   b  and  446   c.  As shown, the first optical gain region  410   a  is pumped with two contra-propagating pump laser beams  404 . 
     In accordance with the Mamyshev architecture, the first and second fiber Bragg gratings  412  and  414  have reflectivity profiles which are spectrally detuned with respect to one another. In this specific embodiment, the first and second fiber Bragg gratings  412  and  414  have refractive index profiles each having a varying grating period and a Gaussian-like apodization, thereby imparting reflectivity profiles of satisfactory reflectivity in terms of maximal reflectivity value and of full width at half maximum bandwidths. However, as mentioned above, the second fiber Bragg grating  414  is only optional, as any other filter of satisfactory filter profile may be used. 
       FIG.  7    shows another example of a fiber laser system  500  in accordance with a ring-type Mamyshev oscillator architecture. As depicted, the fiber laser system  500  has a first pump laser  502   a  and a length  506  of optical fiber which is optically coupled to the first pump laser  502   a  via a first WDM coupler  546   a.  The length  506  of optical fiber has a laser cavity  508  with at least a ring cavity path  509 , a first fiber Bragg grating  512  along the circular cavity path  509 , a second filter  511  along the circular cavity path  509  spaced apart from the first fiber Bragg grating  512 , and a first optical gain region  510   a  between the first fiber Bragg grating  512  and the second filter  511 . An output  516  is optically coupled to the ring-type laser cavity  508  via a 90/10 optical coupler  546 ′, where optical pulses  518  are outputted. In this embodiment, the first fiber Bragg grating  512  has a refractive index profile with varying grating period and Gaussian-like apodization, thereby imparting the sought-after reflectivity profile described herein. 
     As illustrated, an optical circulator  586  is used to circulate optical pulses  518  reflected by the respective first fiber Bragg grating  512  back on a different optical fiber segment, thereby allowing the optical pulses  518  to travel around the ring cavity path  509  in a counter-clock wise manner. In contrast, as the second filter  511  is transmissive, no optical circulator is necessary to maintain the circulation of the optical pulses  518  along the ring cavity path  509 . 
     Also shown in this example, the fiber laser system  500  has a second pump laser  502   b  which is optically coupled to the laser cavity  508  via a second WDM coupler  546   b.  The laser cavity  508  shown in this example also has a second optical gain region  510   b  being pumpable at least by a third pump laser  502   c  which is optically coupled thereto via a third WDM coupler  546   c.  Optical isolators  588  may be provided to block optical pulses propagating in the opposite direction. 
     It is noted that the fiber laser system  500  differs from the fiber laser system  400  of  FIG.  6    in that the second filter  511  is not a fiber Bragg grating offering a reflectivity profile but instead a dielectric filter offering a transmissivity profile. As can be expected from a Mamyshev architecture, the first reflectivity profile of the first fiber Bragg grating  512  is spectrally detuned from a filter profile of the second filter  511 , thereby enabling only optical pulses experiencing spectral broadening to subsist within the laser cavity  508 , as in any Mamyshev oscillator. Other examples of the second filter can be used in some other embodiments. 
     EXAMPLE 1 
     All-Fiber Mamyshev Oscillator Enabled by Chirped Fiber Bragg Gratings 
     Ultrafast fiber lasers thrive with the promise of offering more efficiency, compactness, robustness and simplicity than well-established bulky solid-state mode locked lasers. In the past, for high peak power applications, fiber laser oscillators may have struggled to manage nonlinearity in their tightly confined waveguide medium and could have been more competitive. Recently, a fiber laser system based on a new nonlinear saturable absorber (SA), called the Mamyshev oscillator (MO), was found to be interesting compared to existing lasers in various wavelength regions. It can now offer megawatt peak power levels while benefiting from the robustness of PM fibers. The fiber laser system can rely on self-phase modulation and two detuned spectral filters to support mode locking. Until now, high energy fiber laser system used paired collimators and diffraction gratings to achieve Gaussian filtering. However, these promising performances could benefit from implementation into a cost-effective all-fiber format with a reliable self-starting process to initiate a widespread deployment of their use outside specialized laboratories. The challenge is to obtain efficient and preferably tunable filters in an all-fiber format that are reliable at high power. 
     In this example, there is proposed a linear-cavity MO based on two chirped fiber Bragg gratings (CFBG) with Gaussian apodization as spectral filters. The chirp is preferable in the two FBGs to achieve high reflectivity for the required bandwidth. Moreover, it allows us to manage the nonlinearity in the Mamyshev oscillator by using highly dispersive filters. Here, there is presented an experimental MO-based fiber laser system that generates pulses with an energy of 21.3 nJ and a duration of 108 fs after compression. The results are in agreement with numerical simulations that allow us to propose guidelines for the CFBG-based design and to study the impact of filter dispersion on the nonlinear pulse evolution. 
     At first glance, the fiber laser system  600 , shown in  FIG.  8   , may look like a standard continuous wave fiber (CW) laser. It is entirely made up of PM fibers. A 10.8-m-long small core erbium-doped fiber  610  (EDF07-PM SR from OFS) provides gain and normal dispersion at 1550 nm. At each end of the laser cavity  608 , first and second CFBG  612  and  614  each acts as a ˜5-nm bandwidth reflective filter. The CFBGs  612  and  614  are written in a standard passive fiber (SM15 from Fujikura) with a high level of photosensitivity. The two passive segments have an in-cavity total length of ˜0.60 m. The group delay dispersion (GDD) for a single pass in the laser cavity  608  is estimated at 0.38 ps 2 . 
     Measurements of the first and second reflectivity profiles  622  and  624  along with corresponding group delay dispersion of the CFBGs  612  and  614  are shown in  FIGS.  8 A-B . Their linear chirp and apodization were optimized through modeling to provide a smooth Gaussian reflective profile with minimal sidelobes, a relevant factor for maximizing the output pulse energy and mode locking stability. The CFBGs  612  and  614  have sidelobes at −30 dB (LR-CFBG) and −28 dB (HR-CFBG) relative to their respective maximal reflectivity values R 1  and R 2 . They were UV-written with a chirped phase mask of 100 nm/cm. Very short inscription length with chirped Gaussian apodization (FWHM≅400 μm) and high index modulation (˜3×10 −3 ) are exploited to achieve high reflectivity with large bandwidth in this example. However, other methods of inscription can be used in some other embodiments. For given bandwidth and maximal index modulation, a chirped FBG yields a higher reflectivity than a uniform FBG, as discussed above. Deviation from a perfect Gaussian and asymmetry of the reflectivity profiles  622  and  624  may come from defects in the apodization profile. These small deviations also impact the group delay curves  623  and  625  that, otherwise, would be perfectly linear over the whole bandwidths Δλ. These artifacts were difficult to avoid since accurate inscription is a sizable challenge for sub-millimeter CFBGs with high index contrast. 
     A spectral detuning of 15 nm between the first and second reflectivity profiles  622  and  624  of the CFBGs  612  and  614  allows a spectrally broadened pulse with high peak power to survive in the laser cavity while any CW oscillation is suppressed. The fiber laser system behaves like an ideal nonlinear SA with a large modulation depth that favors high energy pulses. However, it also prevents self-starting from noise in some embodiments. In order to solve this problem, two stretching elements  640   a  set 12.5 cm apart are added to hold and stretch the 1550 nm HR-CFBG  622 , which redshifts the reflectivity curve of the corresponding grating. Once the filters are closer to each other, the SA&#39;s modulation depth is reduced and small noise fluctuations with low peak power can be transmitted through both filters, eventually leading to the formation of high-energy pulses. 
     This behaviour was not observed in this example. Rather, the laser cavity tends to emit CW when the spectral filter wings overlap in this specific embodiment. Thus, an external starting arm  640   b  was added to get reliable self-starting for this experiment. The arm  640   b  uses a saturable absorber mirror (SAM-1550-50-10ps from BATOP GmbH) to induce a noisy Q-switching at 1565 nm with a feedback from the auxiliary output  648 . Once a flip mirror  670  is put down, the laser cavity  608  immediately transitions toward a stable single pulse mode locked regime. At this stage, the stretching elements  640   a  can be used to tune the HR-CFBG  612  back and forth from 1550 to 1560 nm without losing mode locking. 
     Even if this is not essential to the fiber laser system  600  described herein, it adds an interesting degree of freedom. A full characterization of the main output pulse  618  is shown in  FIGS.  9 A- 9 D  along with a comparison with simulation results. The simulations account for self-phase-modulation (SPM), second-order dispersion (GVD), third-order dispersion (TOD) and spectral gain through the scalar generalized nonlinear Schrodinger equation. The measured reflectivity and dispersion profiles are directly used to represent both CFBGs  612  and  614 . The gain is calculated from the erbium cross sections and the rate equations are solved in the steady state. In a linear configuration, the optical pulse  618  passes twice through each section of the optical gain region  610  at each round trip. The transition rates at each position along the fiber are thus computed taking into account the pulse energy as it goes toward the HR-CFBG  612  and the pulse energy of its reflected counterpart going toward the LR-CFBG  614 . 
     A stable 21.3-nJ pulse train at 8.935 MHz is observed on the radio frequency spectrum with an average power of 190 mW. The injected pump power is 850 mW for an excellent 22.3% power conversion efficiency. The pulse energy is limited by the injected pump power. On the secondary output  648 , a pulse of 11.7 nJ is observed. By design, most of the energy is extracted by the pulse co-propagating with the pump laser beam  604 . The main output&#39;s broad spectrum can support transform-limited pulses of 74 fs. The 15 cm segment of anomalous GVD passive fiber at the main output  618  of the laser cavity  608  is partially responsible for the spectral modulations shown in  FIG.  9 A  and helps replenishing the spectral hole left behind by the LR-CFBG  614 . It is interesting to note that no polarizer is required in the laser cavity  608 . The fiber laser system  600  naturally chooses one linear polarization state among the two orthogonal modes of the PM fiber, as can be seen by analyzing the main output  618 . With a polarizer, an extinction ratio of −20 dB was measured. This self-polarizing effect is caused by the advantageous contribution of SPM over cross-phase modulation (XPM) to the spectral broadening. A vectorial version of the numerical model also showed this dynamic even without considering the group delay difference between the fast and slow axis of the fiber. In the experimental setup, the birefringent fiber is essential to prevent any disruption to the polarization state. It also contributes to a greatly improved robustness to environmental perturbations: touching or bending the fiber had no effect on the mode locked state or output pulse characteristics. The pulses were dechirped (or compressed) with a grating pair compressor (600 lines/mm) and the autocorrelation trace is shown in  FIG.  9 B . The PICASSO algorithm was used to get an accurate reconstruction of the pulse envelope. The compressed pulse peak power was estimated to be 95 kW with a FWHM duration of 108 fs. If the 50% loss in the compressor is taken into consideration, the actual experimental peak power is 47.5 kW. In addition to this, a significant fraction of the pulse energy is lost in side lobes spread over 3 ps. This results from a nonlinear, although smooth, variation of the output pulse instantaneous frequency. Potential phase noise coming from the CFBGs is not important here since the 4.7 nm bandwidth cannot strongly affect the 115-nm spanning spectrum after full nonlinear broadening. Excellent agreement with the numerical results provides validations of both the measurements and the numerical model. This gives us the opportunity of further exploring the pulse dynamics inside the laser cavity  608  and the impact of the GDD of the CFBGs  612  and  614 . 
     Through simulations, the impact of the dispersion of the HR-CFBG  612  on the laser cavity  608  was studied over a wide range of GDD and for two values of reflectivity of the LR-CFBG  614 . For this analysis, both filters&#39; reflectivity parameters are considered to be perfectly Gaussian with 4.5 nm bandwidth. The passive fiber after the LR-CFBG  614  at the main output  618  is not taken into account. All results are taken when the laser cavity  608  reaches the steady state.  FIGS.  10 A- 10 B  show the pulse energy and its peak power after optimal compression via compensation of second-order dispersion as a function of the HR-CFBG&#39;s GDD.  FIG.  10 A  is essentially the same as this experimental setup while case 
       FIG.  10 B  shows the impact of a lower LR-CFBG reflectivity without dispersion. It can be appreciated that as long as the GDD increases, in either the normal dispersion regime or the anomalous dispersion region, the pulse energy also increases. 
     In steady state, the main pulse and its reflected counterpart share the same local gain along the fiber. For that reason, both of their nonlinear evolution and amplification process are coupled to each other through the longitudinal gain profile. To demonstrate this effect,  FIG.  10 A  shows that a lower LR-CFBG reflectivity without dispersion could give a significant improvement on the main pulse energy by giving it less competition on the energy extraction and by cutting a lesser portion of the spectrum. However, the filter dispersion does not cause a loss of energy inside the cavity and only slightly affects the pulse energy. This small perturbation results from a modified nonlinear spectrum broadening which affects the wavelength-dependent gain along the fiber. 
     The cavity can support higher dispersion in the negative GDD region. With the 41% LR-CFBG, the difference is 0.5 ps 2  which is similar to the single pass GDD. When negative, the CFBG dispersion has to compensate for the positive chirp accumulated along the fiber. This means a weaker inhibition of the initial peak power and pulse&#39;s nonlinear broadening for a higher anomalous dispersion. For the same reason, a cavity with lower LR-CFBG reflectivity will tolerate smaller maximum filter dispersion. 
     Higher peak power after compression indicates a more linear initial chirp and a smoother spectrum leading to a lower energy lost in the sidelobes of the compressed pulse. In this example, the simulation shows that a positive GDD value of +0.55 ps 2  is optimal for the output pulse compressibility. In practice, with such a low GDD, CFBGs with good properties could not be achieved since there is a limitation in terms of maximum index modulation. More dispersion means that a higher peak reflectivity can be achieved by the inscription of a longer CFBG with a smaller chirp for the same maximum refractive index modulation and bandwidth. Thus, it is operated in a higher optimal anomalous GDD in order obtain the necessary bandwidth and reflectivity to keep a high pulse energy. Operating the CFBG with a higher negative GDD has other positive side effects. First, the pulse avoids losses from coupling with the fiber&#39;s cladding modes that can result from operating in positive GDD. Secondly, simulation indicates that higher dispersion can mitigate the impact of asymmetries in the filter shape that are bound to arise from imperfections in the inscription. Overall, the results of  FIGS.  10 A- 10 B  suggest the possibility of tailoring the nonlinear pulse evolution without a major effect on the output pulse energy. The general trend to remember is that a moderate amount of filter dispersion (positive or negative) is beneficial. To further understand the underlying dynamics behind this, simulation results in  FIGS.  11 A- 11 C  compare the pulse evolution inside the gain fiber between the worst and the optimal HR-CFBG GDD values. The solid lines show the evolution of the pulse parabolic misfit parameter, the pulse energy, and the pulse bandwidth when chirped fiber Bragg gratings are used. The dashed lines show the evolution of those parameters when the fiber Bragg gratings are not chirped for comparison. 
     It is interesting to notice that the pulse energy grows in a similar manner for both cases. Even with significant manipulation of the nonlinear dynamics, the filter dispersion does not strongly affect the pulse energy because the saturated gain distribution in the steady state will roughly remain the same for a fixed pump power and reflectivities of the CFBGs. The evolution of the misfit parameter M shows two different nonlinear regimes. First, the pulse parameters follow an exponential growth (energy, RMS bandwidth, etc.) until a minimal M value is reached. This is typical for a self-similar (SS) regime which attracts any input pulse toward a parabolic shape. Secondly, the spectrum broadens and eventually hits the gain bandwidth limit which stops the nonlinear attractor. The pulse shifts into a simple SPM and gain regime where the linear chirp starts to bend as the pulse slowly loses his parabolic shape due to the gain bandwidth effect. This state has a similar behaviour than the transitional state between the self-similar and gain-managed nonlinearity (GMN) regimes since the blue part of the broadened spectrum undergoes some absorption. This is expressed by a saturation of the pulse spectrum RMS bandwidth toward the fiber end. Stretching the pulse before amplification reduces the initial peak power and slows down the spectral broadening. As a result, the inflection point between the two regimes is pushed further along the fiber. By reducing the SPM amplification regime length of action, the output pulse chirp is more linear. On the other hand, it is expected to observe a slower rate of convergence to the parabolic shape for an initial pulse with higher chirp and duration. This also results in a spectrum which is not as smooth in most of the amplifier length. If this effect becomes too strong, the smooth nonlinear evolution brought by the self-similar regime cannot take place. Prohibited from achieving a parabolic shape, the pulse is undermined by optical wave breaking and the cavity becomes unstable. This explains the dispersion upper-limits observed in  FIGS.  10 A- 10 B . The best GDD value gives the optimal trade-off between the bandwidth effect and weaker self-similar regime. 
     To conclude this example, an all-fiber Mamyshev oscillator is proposed which emits 21.3-nJ pulses compressible to 108 fs with an efficiency of 22.3% relative to the launched pump power. This achievement was made possible through the use of sub-millimeter Gaussian-shaped and chirped fiber Bragg gratings as spectral filters at the extremities of a linear cavity. The chirp of the gratings is essential to be able to fabricate high-reflectivity Gaussian filters with the bandwidth required in an MO. The chirp also provides the opportunity of controlling the dispersion of the filters, which has an important impact on the nonlinear evolution of the pulse in an MO. It was shown in this example that an amount of filter GDD comparable to the fiber GDD is beneficial to obtain larger pulse peak power without affecting the pulse energy. This might be a first step toward more complex pulse shaping schemes in Mamyshev oscillators. Arguably, this is the simplest ultrafast fiber laser architecture so far and still it yielded competitive results in terms of pulse energy, duration and peak power. This paves the way to accessibility and cost effectiveness which will enable many applications requiring high-energy pulses. Since it is compatible with high power requirements, this approach also shows potential to bring the MW power level of the ytterbium-based fiber MO to an all-fiber format. 
     In the following examples, the index modulation used in the grating inscription process is given by the following equation: 
         n ( z )= n   0 +max( A )+ A ( z ) cos( z /Λ( z )),   (1)
 
     where z represents the longitudinal position along the fiber, A is the apodization profile, n o  is the initial index of the fiber and Λ is the grating period. 
     EXAMPLE 2 
     Fiber Bragg Grating Operating at 1552 nm in an Erbium-Based Laser Cavity 
       FIG.  12 A  shows an example of a fiber Bragg grating designed to obtain Gaussian-like reflectivity profile near about 1552 nm for an erbium-based laser cavity. In this specific example, the grating is inscribed with specific apodization and a linear chirped phase mask period of 65 nm/cm. Measured grating was inscribed in the fiber PM980-XP from Nufern.  FIG.  12 B  shows the measured reflectivity profile of that fiber Bragg grating. In this specific example, the length of the fiber Bragg grating along the z-axis is about 2000 μm, the apodization profile A that is Gaussian-like as shown at  FIG.  12 A , the initial index n o  is 1.44897, the grating period is given by Λ(z)=0.362 um+z*(32.5 nm/cm), where z denotes the center of the apodization profile. These specific values are given as examples only and are non-limiting as they can differ from one embodiment to another. 
     EXAMPLE 3 
     Fiber Bragg Grating Operating at 1050 nm in an Ytterbium-Based Laser Cavity 
       FIG.  13 A  shows another example of a fiber Bragg grating. In this specific example, the fiber Bragg grating is designed to exhibit a Gaussian-like reflectivity profile near about 1050 nm for an ytterbium-based laser cavity. The grating is inscribed with specific apodization and a uniform grating period, and was inscribed in the fiber Liekki Passive-10/125DC-PM from nLIGHT.  FIG.  13 B  shows the measured reflectivity profile of that fiber Bragg grating. In this specific example, the length of the fiber Bragg grating along the z-axis is about 700 μm, the apodization profile A that is Gaussian-like as shown at  FIG.  13 A , the initial index n o  is 1.45201, the grating period Λ is 0.535 μm. These specific values are given as examples only and are non-limiting as they can differ from one embodiment to another. 
     As can be understood, the examples described above and illustrated are intended to be exemplary only. For instance, the first fiber Bragg grating may not be a fiber Bragg grating in some embodiments. In these embodiments, the laser system can have a pump laser, and a laser cavity optically coupled to the pump laser. The first fiber Bragg grating may have a uniform grating period in some embodiments. In some embodiments, the laser cavity may be fibered. The laser cavity has a cavity path, a first filter having a first filter profile of a Gaussian-like shape, a second filter having a second filter profile, and an optical gain region between the first and second filters along the cavity path. As in any Mamyshev oscillator, the first and second filter profiles are spectrally detuned from one another. However, in this embodiment, the first filter is a dispersive filter thereby imparting a dispersive profile across at least at portion of the first filter profile. As such, when the optical gain region is pumped with the pump laser, and when laser cavity is mode locked, optical pulses of satisfactory pulse energy and compressibility quality can circulate within the laser cavity, and then outside the cavity via an output. In these embodiments, the first filter can be a dispersive filter of any suitable type including, but not limited to, Bragg filters, volume Bragg filters, fiber Bragg gratings, three dimensional Bragg filters and the like. The scope is indicated by the appended claims.