Abstract

A significant change in active similariton characteristics, both numerically and experimentally, is observed as a function of the location of the lumped spectral filter. The closer the spectral filter is to the input of the Yb3+-doped fiber, the shorter the de-chirped pulse width. The peak power of the de-chirped pulse has its maximum value at a certain location of the spectral filter. Four different positions of the spectral filter inside the laser cavity have been theoretically studied and two of them have been verified experimentally.

© 2015 Optical Society of America

1. Introduction

Self-similar (similariton) pulses with parabolic temporal profiles and linear frequency chirp have many attractive features. This includes great immunity against optical wave breaking [1, 2], transform-limited compression ability and an ability to carry high energy levels.

Similariton generation can be classified into two categories; active and passive similaritons. This classification arises from the dynamics of the pulse evolution inside the laser cavity. M. E. Fermann et al. reported the conversion of any arbitrary pulse profile into a parabolic pulse (similariton) with linear frequency chirp after its propagation through a long-enough gain (active) medium having normal dispersion. This conversion is due to the local nonlinear attractor in the gain medium, which forces the pulses to evolve towards a parabolic profile [1]. Hence, an active similariton is formed through the gain medium inside the laser cavity, while a passive similariton depends on all the parameters of the laser cavity. Ilday et. al. demonstrated passive similariton generation from dispersion-managed (DM) Yb3+-doped fiber laser cavity employing gratings for anomalous dispersion [3]. An all-normal dispersion (ANDi) cavity has paved the way toward an all-fiber Yb3+-doped laser cavity. Since ANDi fiber laser is free of anomalous dispersion components, a spectral filter (SF) is an essential element to compensate the pulse (temporal and spectral) broadening during evolution inside the laser cavity. ANDi became a platform for a new pulse regime called a dissipative soliton as well as an active similariton. Based on the bandwidth of the SF and the length of the gain medium, ANDi fiber laser can generate either a dissipative soliton or similariton. The insertion of narrow bandwidth SF (2 nm - 4 nm) assists the generation of an active similariton [4–6]. Wider SF in an ANDi cavity produces dissipative solitons [7, 8]. In previous work of the authors, it was found that the positioning of the wide SF in the laser cavity has no significant effect on the characteristics of the dissipative soliton [9].

In this paper, the variation of the active similariton characteristics (spectral bandwidth, peak power and de-chirped pulse width) based on the location of the SF is investigated theoretically and experimentally. Average models cannot be used to model active similariton lasers because the pulse evolution relies on a local nonlinear attractor in the Yb3+-doped fiber, which forces the pulses to evolve towards a parabolic profile [3, 4]. Therefore, the Ginzburg-Landau equation, whether in its scalar form or vector form, is used to simulate the pulse propagation in the Yb+3-doped fiber and single mode fibers (SMFs). To the best of our knowledge, the position of lumped SF in active similariton fiber laser has not been investigated before.

The paper is organized as follows: the mode-locked laser cavity is described in section 2. In section 3, the simulation model is illustrated. Discussion of the simulation results appears in section 4. The experimental results are demonstrated in section 5. Finally, the conclusion is presented.

2. Laser cavity description

Figure 1 shows the mode-locked laser cavity designed to investigate the proposed idea. It is mode-locked by nonlinear polarization rotation (NPR). Two quarter wave plates (QWPs), one half wave plate (HWP) and a polarization beam splitter (PBS) are inserted inside the cavity to initiate the mode-locking.

 figure: Fig. 1

Fig. 1 Schematic diagram of the active similariton laser cavity having the SF at four positions to test its effect on the temporal and spectral properties of the pulse; QWP is the quarter wave plate, HWP is the half wave plate and ISO is the Faraday isolator.

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Four cases are illustrated to study the effect of the position of the lumped SF. For all cases, the total cavity length is equal and its value is 8.74 m. The lengths of the Yb3+-doped fiber (Lyb) and the following SMF (LSMF2) are unchanged in all cases. They are taken to be 1.98 m and 0.4 m, respectively. In addition, the whole length of SMF (6.36 m) between the input of the Yb3+-doped fiber and QWP2 is kept constant. For Case 1, the SF is located right after QWP2, while, for Case 2, the SF is located at 1.6 m away from QWP2. For Case 3, the SF is inserted at 4.53 m away from QWP2. Finally, for Case 4, the SF is positioned at the input of the Yb3+-doped fiber.

It is worth mentioning that the experimental implementation of Case 1 and Case 4 is difficult because of the leads of the wave division multiplexer (WDM) and the SF; the lengths of the input and output fiber leads are 0.6 m and 0.3 m for the WDM and the SF, respectively. Also, the length of the SMF (0.4 m) attached to the collimator after QWP2 is added to the length of the input fiber lead of the SF that makes the implementation of Case 1 impossible. Furthermore, if the SF is inserted between the WDM and the input of the Yb3+-doped fiber, as for Case 4, it will not pass the pump light, which is at 980 nm, to the Yb3+-doped fiber. Therefore, only Case 2 and Case 3 are experimentally constructed. To match with the experimental set-up, a coupler of output coupling ratio 0.05 is inserted at 0.81 m from QWP2 in the simulation model for Case 2 and Case 3. The main output of the mode-locked laser cavity is at the rejection port of the PBS, which is named Output 1 throughout this paper.

3. Simulation model parameters

The evolution of the optical pulse in Yb3+-doped fiber and SMFs is modeled by vector coupled Ginzburg-Landau equations (CGLEs) given as [10]:

uz+iβ222ut2β363ut3+αf2u=g(z)2(1+1Ωg22t2)u+iγ[|u|2+A|v|2]u+iγBu*v2.
vz+iβ222vt2β363vt3+αf2v=g(z)2(1+1Ωg22t2)v+iγ[|v|2+A|u|2]v+iγBv*u2.
where u and v represent the two orthogonally polarized electric field envelopes; the z axis denotes the propagation distance through the fiber; t is the retarded time; γ is the nonlinear parameter; it is taken as 0.005 W−1m−1 and 0.0041 W−1m−1 for the SMFs and Yb3+-doped fiber, respectively; A = 2/3 and B = 1/3 for silicate fibers; Ωg = 40 nm is the gain bandwidth of the Yb3+-doped fiber; β2 and β3 are the group velocity dispersion (GVD) and third order dispersion (TOD). Their values at 1030 nm are 24690.1 fs2/m and 75.3 fs3/m for the SMFs, and 22752.5 fs2/m and 74.2 fs3/m for Yb3+-doped fiber, respectively. αf is the linear power loss coefficient.

The gain in the Yb3+-doped fiber is expressed as:

g(z)=go1+((|a|2)/Esat).
Here go is the small signal power gain coefficient taken as 17.5 dB/m.|a|=|u|2+|v|2  is the magnitude of the total field. Esat is the saturation energy of the Yb3+-doped fiber. It is adjusted for each case to have stable single-pulse operation. For Case 1 and Case 4, Esat has a value of 1.25 nJ. To account for the coupling loss in Case 2 and Case 3, Esat has a slightly higher value of 1.3 nJ. The gain term is omitted for the pulse propagation inside the SMFs.

The lumped SF is modeled as a Gaussian filter with insertion loss 0 dB and 3-dB bandwidth of 3.2 nm. Fast Fourier transform is applied to the pulse at the input of the SF. Then, the pulse in the frequency domain is multiplied by the transfer function of the SF. Finally, inverse Fourier transform is applied to the output spectral components to retrieve the pulse back to the time domain.

To have a fair comparison, the orientation angles of the wave plates are kept constant for all cases. Their values are selected as follows: the orientation angles of QWP1, HWP and QWP2 are 121.5°, 151° and 43.65°. The through and rejection ports of the PBS, the wave-plates and the Faraday isolator are modeled by their Jones matrices [11, 12]. The simulation model is initiated by a small pulse of unity amplitude. 1500 roundtrips were sufficient to have a stable single-pulse mode-locked operation.

4. Simulation results and discussion

For all cases, the normalized temporal profiles of the chirped pulses at the end of LSMF2, the through port of the PBS and Output 1 are shown in Figs. 2(a)-2(d). The excess kurtosis factor is used to verify if the pulse approaches the parabolic profile. It has a value of −1.2, −0.86, 0 and 1.2 for rectangular pulse, ideal parabola, Gaussian, and Sech2 profiles, respectively [13]. They are reported in Table 1 at various locations inside the mode-locked laser cavity.

 figure: Fig. 2

Fig. 2 The numerical normalized temporal profile of the chirped pulse at the end of LSMF2 (SMF2), the through port of the PBS (Pol), the rejection port of the PBS (Output 1) and the coupler output: (a) Case 1, (b) Case 2, (c) Case3, and (d) Case 4. (e) Plot of the FWHM of the chirped pulse

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Tables Icon

Table 1. Excess Kurtosis factor of the chirped pulses

These values show that the chirped pulses for the four cases are close to having a parabolic temporal profile only at the end of LSMF2. This can be explained as follows: the long length of the Yb3+-doped fiber has the capability to convert the input pulse into a parabolic one [1]. However, the through port of the PBS is affected by the nonlinear transmission of the NPR that filters out the low-power edges of the pulse and converts it to the shape shown in Figs. 2(a)-2(d). In contrast, the low-power portions of the pulse at the end of LSMF2 that contain the sharp edges are passed through Output 1. Then, the temporal profile of the pulse tends to have a rectangular shape.

As shown in Fig. 2(c), the chirped pulse at the coupler output for Case 3 has some distortion at its edges that is not seen in Case 2 (see Fig. 2(b)). It can be explained as follows: the peak power of the pulse at the through port of the PBS for Case 3 is 2.66 times higher than the corresponding peak power in Case 2. This factor enhances the interplay between the dispersion and the self-phase modulation, which results in temporal distortion and an interference structure in the spectral profile.

The full width at half maximum (FWHM) of the temporal profile of the chirped pulses at the end of LSMF2, through the port of the PBS and Output 1 is plotted in Fig. 2(e). It is clear that the FWHM decreases with the proximity of the lumped SF to the input of LYb. For Case 2 and Case 3, the FWHM was calculated to be 4.28 ps and 3.45 ps at Output 1, respectively.

Figure 3 shows the spectral power densities (SPDs) of the pulses at the end of LSMF2, the through port of the PBS and Output 1 for all cases. The SPDs of the pulses at the coupler output for Case 2 and Case 3 are shown as well. The SPDs of the pulses do not have any sharp edges, which is a signature of having active similariton mode-locking [4]. The SPDs of Case 3 and Case 4 have more oscillatory structures compared to the corresponding SPDs of Case 1 and 2. The evolution of the peak power of the pulse through the cavity is plotted in Fig. 4(a). The input of the Yb3+-doped fiber is taken as the start point for each case. It should be noted that the wave plates, PBS, Faraday isolator and SF are represented by virtual lengths in Fig. 4(a) to depict the evolution of the pulse peak power through these components. The higher values of peak power at the end of LSMF2 enhance the effect of SPM that causes spectral broadening along with these oscillatory structures. As shown in Fig. 4(b), the SPD of the chirped pulses is broadened with the proximity of the lumped SF to the input of LYb. For Case 2 and Case 3, the spectral bandwidths were calculated to be 11.38 nm and 20.1 nm, respectively.

 figure: Fig. 3

Fig. 3 The numerical normalized SPD of the chirped pulse at the end of LSMF2 (SMF2), the through port of the PBS (Pol), Output 1 and the coupler output: (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4.

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 figure: Fig. 4

Fig. 4 (a) Evolution of the total peak power through the four mode-locked laser cavities. (b) Spectral bandwidth at various locations inside the cavity. The locations of LYb, LSMF2, wave plates and polarizer are shown for one of the cases (Case 4) for the sake of clarity. 1: QWP1; 2: HWP; 3: the through port of the PBS; 4: Faraday isolator; 5: QWP2; 6: Lumped SF.

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For Case 2 and Case 3, the autocorrelation (AC) profiles of the chirped pulses at Output 1 are plotted in Fig. 5(a) to be compared with its measured counterpart. The AC deconvolution factors were calculated to be 0.93 and 0.96, respectively.

 figure: Fig. 5

Fig. 5 (a) The normalized AC of the chirped pulse at Output 1 for Case 2 and Case 3. Temporal profile of the de-chirped pulses: (b) Case 1 and Case 2 and (c) Case 3 and Case 4. Normalized AC of the de-chirped pulses: (d) Case 1 and Case 2 and (e) Case 3 and Case 4.

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For all cases, the chirped pulses at Output 1 were passed through a linear dispersion delay line (no nonlinearity is included). It provides the required anomalous dispersion to compress the pulse near its transform limit. We scanned the anomalous dispersion values over a wide range till reaching the minimum value of the FWHM of the pulse. The temporal profiles of de-chirped pulses are shown in Figs. 5(b) and 5(c). The AC profiles of these pulses are shown in Figs. 5(d) and 5(e). The AC deconvolution factor was calculated to be on average 0.75. This indicates that the temporal profile of the pulse is close to having a Gaussian shape (the AC deconvolution factor of a Gaussian pulse is 0.709). The FWHM of the temporal profile of the de-chirped pulse is plotted in Fig. 6(a). The pulse is shortened with the proximity of the lumped SF to the input of LYb, which is logical as the spectral bandwidth increases in the same direction.

 figure: Fig. 6

Fig. 6 (a) Plot of FWHM of the de-chirped pulse. (b) Plot of peak power of the chirped and de-chirped pulses.

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The peak power of the chirped and de-chirped pulses is plotted in Fig. 6(b). The peak power of the chirped pulse changes slightly from one case to another. However, the peak power of the de-chirped pulse has its highest value in Case 3. Then, it decreases again at Case 4. If the pulse energy is the same for each case, the peak power of the de-chirped pulse should increase because of the reduction of the pulse width. However, the simulation results show a monotonic reduction of the pulse energy with the lumped SF approaching the input of LYb (see Fig. 7(a)). Furthermore, as shown in Fig. 3, the SPD of Case 4 depicts an oscillating structure, which leads to having the highest percentage of energy in the side pulses (see Fig. 7(b)).

 figure: Fig. 7

Fig. 7 Plot of (a) pulse energy at Output 1 and (b) percentage of the pulse energy of the side pulses to the total energy.

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To have a clear understanding of the simulation results, the evolution of the root mean square (RMS) temporal and spectral widths for the u-field (horizontally polarized light) and the v-field (vertically polarized light) through the four cases of mode-locked laser cavities are plotted in Fig. 8. For all cases, the lumped SF reshapes the chirped pulse to have approximately the same value of RMS for temporal and spectral widths and peak power.

 figure: Fig. 8

Fig. 8 Evolution of the spectral and temporal RMS widths through the four mode-locked laser cavities: (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4. The solid lines represent the u-field, while the dashed lines represent the v-field. 1: QWP1; 2 HWP; 3: the through port of the PBS; 4: Faraday isolator; 5: QWP2; 6: Lumped SF. Virtual lengths are set to these components for the sake of illustration.

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For Case 1 and Case 2, the pulse propagates through a long length of SMF (LSMF1) (6.36 m or 4.76 m, respectively) that leads to an increase in RMS temporal and spectral widths due to dispersion and nonlinear effects. Therefore, the RMS temporal width has the highest value at end of LSMF2. However, as shown in Figs. 8(a) and 8(b), the RMS spectral width tends to saturate at the middle of LSMF1 because of the reduction of the pulse peak power (see Fig. 4(a)).

As shown in Figs. 8(b)-8(d), the RMS temporal and spectral widths increase through the SMF after QWP2 (LSMF3). However, the RMS spectral width for Case 3 and Case 4 follows a saturation behavior because of the reduction of the pulse peak power (see Fig. 4(a)) that results from the dispersion effect.

For Case 3 and Case 4, the pulse propagates through a short length of SMF (LSMF1 = 1.825 m) or directly through the Yb3+-doped fiber, respectively. The pulse peak power at the beginning of the Yb3+-doped fiber is higher than the corresponding values in Case 1 and Case 2 (see Fig. 4(a)). Because of the self-phase modulation, the SPD of Case 4 has the broadest spectral bandwidth with deep oscillations. No sign of wave breaking was seen in either Case 3 or Case 4 through the Yb3+-doped fiber and LSMF2 because of the linearization of the frequency chirp produced by the conversion of the pulse to similariton.

To explain the reason behind the decrease of the pulse energy from Case 1 to Case 4 (see Fig. 7(a)), we can define two sources of energy loss inside each laser cavity; the SF loss and the loss induced by the rejection port of the PBS. The temporal and spectral dynamics of the chirped pulse are affected by the SF location. The SF produces an energy loss to the chirped pulse because of the spectral and temporal filtering. The value of energy loss for each case is reported in Table 2. It is clear that the SF produces the lowest value of energy loss in Case 1, while the highest value is for Case 4. As shown in Fig. 8(a), the RMS temporal and spectral widths of the chirped pulse have the lowest values at the input of the SF in Case 1. In contrast, they have the highest value in Case 4 (see Fig. 8(d)). As shown in Table 2, the pulse energy for each case has approximately the same value at the end of LSMF2 and after the SF. This means that the value of the saturated gain is approximately the same. As the net gain of any oscillator should equal zero at steady state, the amount of energy loss induced by the rejection port of the PBS changes in the cavity to satisfy this condition. As a result, Case 1 has the highest value of loss at Output 1, which is interpreted as the highest value of output energy. As the value of SF loss increases, the value of energy loss at Output 1 decreases, which corresponds to the reduction of the output pulse energy.

Tables Icon

Table 2. Chirped pulse energy at the end of LSMF2 and SF loss for each laser cavity

Spectral filtering plays a critical role in stabilizing mode-locked all normal dispersion fiber lasers. As an optical pulse propagates in a normal dispersion fiber, it acquires a positive frequency chirp due to dispersion and self-phase modulation. Its pulse width also broadens with the propagation distance. Spectral filtering of the chirped pulse in the frequency domain is mapped into temporal cutting of the pulse edges [14]. The bandwidth of the SF and the nonlinear transmission of the saturable absorber have a major effect on the temporal and spectral profiles of the pulse in the dissipative soliton regime [8, 15–18]. In addition to pulse stabilization, the SF assists the generation of a parabolic pulse in the active similariton laser by reducing the temporal width of the pulse entering the Yb3+-fiber. M. E. Fermann et al. reported the reduction of the length of the gain fiber (Yb3+-fiber) required to fully convert any arbitrary pulse into parabolic pulse with the reduction of the input pulse width [1]. Moreover, we also find that location of the SF has distinct effect on the pulse characteristics.

For all cases, the effect of increasing Esat on the temporal profile of the pulse was studied. We found a distortion in the temporal profile of the pulse at the rejection port of the PBS (Output 1). It appears as a dip in the center of the pulse. By further increasing Esat, as shown in Fig. 9(a), the pulse will split into two large pulses at the edges and some satellite pulses in the middle. Although the laser cavity supports only one pulse per round trip, the AC profile appears as having multi-pulse operation, which is due to the pulse distortion (see Fig. 9(b)). In the same time, as shown in Fig. 9(a), the pulse at the through port of the PBS does not suffer from the same type of distortion and the AC profile shows single pulse operation. The main reason for this pulse distortion, as reported in [19], is due to the nonlinear transmission of the NPR that depends on the selection of the angles of the wave-plates. If the modulation depth of the nonlinear transmission of the NPR is high, the pulse, at Output 1, will have higher amplitude at its edges than at its central region; this is the onset of pulse distortion. It was found that the value of Esat corresponding to the onset of pulse distortion decreases with the proximity of lumped SF to the input of the Yb3+-doped fiber. That is due to the increase of pulse peak power as the filter approaches the input of the Yb3+-doped fiber (see Fig. 4(a)). At much higher values of Esat, unstable pulse operation appears like the results reported in [20]. Unlike [20], we did not see any multi-pulse operation afterward. We believe that the occurrence of multi-pulse operation mainly depends on the nonlinear transmission of the NPR. More detailed study on the effect of the nonlinear transmission of the NPR on the selection of the bandwidth of the lumped SF and multi-pulse operation is now under investigation.

 figure: Fig. 9

Fig. 9 (a) The numerical normalized temporal profile of the chirped pulse at the through port of the PBS (Pol) and the rejection port of the PBS (Output 1) for Case 2 at Esat equals 6.85 nJ. (b) The corresponding AC profiles at the same value of Esat.

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It is worth mentioning that our idea is not directly related to the type of saturable absorber whether it is a real (lumped) or artificial (NPR-based) saturable absorber. However, we reported a direct relation between the bandwidth of the SF and the parameters of the saturable absorber [18]. The bandwidth of the SF must be greater than BWth to have stable mode-locking, which is given as:

BWth=22π1L[(1ζ)β2a1b1γ(l+a1)Ωg2].
where L is the total cavity length. a1 is the small signal saturable loss coefficient, which is related to the saturable absorber modulation depth. l includes the non-saturable loss, the power splitter and filter insertion loss. b1 represents the inverse of the saturation power of the saturable absorber. ζ=c1/a1b1 represents the normalized overdriving term of the saturable absorber. c1 represents the effect of overdriving the saturable absorber. Since an active similariton requires narrow SF bandwidth (2-4 nm), this puts some restriction on the values of the parameters of the saturable absorber.

Two numerical examples are performed to illustrate the effect of the saturable absorber parameters on BWth. X. Liu et al. reported the implementation of a saturable absorber based on single wall carbon nanotube. Its modulation depth and saturation intensity are 12.05% and 9.67 MW/cm2, respectively [21]. The value of non-saturable loss is taken to be 0.42 according to the data reported in [22]. The values of a1, b1 were calculated based on mathematical analysis similar to that in [18] to be 0.013 and 0.15, respectively. The value of ζ is taken to be zero. β2 is averaged over the cavity length. BWth was calculated based on Eq. (3) to be 2.76 nm. This value is inside the proper range of SF bandwidth (2 nm-4 nm) that should be inserted to generate active similariton from the laser cavity [4–6]. This means that, active similariton is a possible mode-locking regime with this specification of the saturable absorber. However, if the saturation intensity of the saturable absorber is increased as reported in [23], the value of BWth will be increased to be 4.83 nm. Therefore, the generation of active similariton will be quite difficult because the bandwidth of the SF is out of the required range in this case.

5. Experimental results

The laser cavity shown in Fig. 1 was experimentally implemented for Case 2 and Case 3 using the same fiber lengths depicted in section 2. An inline fiber SF was purchased from Fiber Logix (its part number is FL-BDF-1030-02-10-N-B-1). Its experimental transfer function is plotted in Fig. 10(a). It has the same 3-dB bandwidth (3.2 nm) as the SF in the simulation model. A light emitting diode centered at wavelength 1030 nm was used to characterize the filter.

 figure: Fig. 10

Fig. 10 (a) Measured normalized transfer function of the lumped SF. (b) Schematic diagram of the diffraction gratings pair combined with a retro-reflector mirror.

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A lightly-doped Yb3+ fiber (its part number is SCF-YB25-6/125-13) purchased from CorActive is used as the gain medium; its core radius is 3 µm; its doping concentration is 1.28 × 1025 ion/m3; its core absorption at 915 nm is 30 dB/m. The optical signal is coupled to/from the free space through angle-cleaved collimators. A free-space Faraday isolator (OFR, IO-5-1030 HP) is inserted inside the cavity for unidirectional propagation. The pulse at Output 1 is de-chirped outside the cavity using a pair of diffraction gratings serving as a linear dispersion delay line. Each diffraction grating has density of 600 lines/mm. Figure 10(b) illustrates the schematic diagram of the diffraction gratings block. A retro-reflector mirror is used to reflect the diffracted beams back to the diffraction gratings at a higher level. The created anomalous dispersion is a function of the separation between the gratings [24].

To ensure a single-pulse operation, the output chirped pulse is monitored using a (120 ps scanning range) Autocorrelator, high-resolution optical spectrum analyzer (0.05 nm), and fast optical photodetector (30 GHz) attached to a sampling-based oscilloscope of the same bandwidth.

For Case 2, the pump power for single pulse operation was found to be 400 mW. The repetition rate of the pulses was measured to be 24.5 MHz. The average power at Output 1 was measured to be 136 mW. Therefore the pulse energy was calculated to be 5.55 nJ. The SPDs of the mode-locked pulse at Output 1 and the coupler output are plotted in Fig. 11(a). The SPDs do not have sharp edges, as seen in Fig. 3(b). It is clear that the SPD at Output 1 has more structure than the corresponding SPD at the coupler output, which is also observed in the simulation results. The spectral bandwidth of the SPD at output 1 has a value of 12.18 nm, while it is 11.88 nm at the coupler output. The values are in good agreement with the simulation results.

 figure: Fig. 11

Fig. 11 Experimental mode-locking SPD at Output 1 (blue solid line) and the coupler output (red dashed line) for (a) Case 2 and (b) Case 3. Experimental AC profile of (c) the chirped pulse and (d) de-chirped pulse at Output 1. Case 2: blue solid line and Case 3: red dashed line.

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The AC profiles of the chirped and de-chirped pulses at Output 1 are plotted in Figs. 11(c) and 11(d), respectively. The values of the FWHM of the AC profiles of the chirped and de-chirped pulses were measured to be 2.87 ps and 194 fs, respectively. The deconvolution factor calculated in the simulation model is used to get the values of the FWHM of the temporal profiles of the chirped and de-chirped pulses to be 2.67 ps and 144.14 fs.

For Case 3, the pump power for single-pulse operation and the repetition rate were approximately the same (400 mW and 24.5 MHz, respectively). The average power at Output 1 was measured to be 116.56 mW. Therefore the pulse energy was calculated to be 4.76 nJ. The pulse energy is lower than its value in Case 2, which matches the simulation results. The SPD of the mode-locked pulse at Output 1 and the coupler output are plotted in Fig. 11(b). Similar to the SPDs plotted in Fig. 3(c), the spectral edges are not so sharp. The spectral bandwidth was measured to be 21.3 nm and 22.47 nm for both outputs, respectively. The values of the spectral bandwidth are close to the numerical results. The spectral profiles at both outputs have structures similar to the simulation results.

The AC profiles of the chirped and de-chirped pulses at Output 1 are plotted in Figs. 11(c) and 11(d), respectively. The values of the FWHM of the AC profiles of the chirped and de-chirped pulses were measured to be 2.58 ps and 148 fs, respectively. Using the deconvolution factor calculated in the simulation model, the values of the FWHM of the temporal profiles of the chirped and de-chirped pulses are 2.48 ps and 109.37 fs. Based on the relation shown in [25], the peak power of the de-chirped pulse was calculated to be 35.2 KW and 40.82 KW for Case 2 and Case 3, respectively.

Based on the results shown in Fig. 11, we found that the time-bandwidth products for the chirped pulses were calculated to be 9.2 and 14.9 for the Case 2 and Case 3, respectively, which is larger than the transform limited product indicating that the pulse is highly chirped. However, for the de-chirped pulse, the value of the time-bandwidth product for Case 3 was 0.66 that is larger than the value for Case 2 (0.5). This can be explained as follows; the higher effect of SPM in Case 3 (due to the higher peak power) (see Fig. 4(a)) reduces the linearity of the frequency chirp of the pulse. This prohibits the linear dispersion delay line to compress the pulse very close to its transform limit.

The experimental values of the FWHM of the chirped and de-chirped pulses have the same qualitative behavior with the variation of the position of the SF within the cavity. Both the simulation and experimental results demonstrate that the position of the SF plays a significant role in the energy and the peak power of the de-chirped pulses. In applications where high pulse energy is required, it is advantageous to insert lumped SF close to QWP2 (Case 1), in which case, the pulse will have higher output energy (see Fig. 7(a)) with reduced value of the spectral bandwidth (see Fig. 4(b)) and wider de-chirped pulse width (see Fig. 6(a)). In contrast, if a high peak power of de-chirped pulses is required, and the lumped SF is located as in Case 3, the de-chirped pulse will have higher peak power (see Fig. 6(b)) and spectral bandwidth (see Fig. 4(b)) and narrower pulse width (see Fig. 6(a)) but with lower value of energy (see Fig. 7(a)).

To confirm that change in the energy and the peak power with the SF location was indeed due to the SF location and not due to the specific cavity length, the total length of the cavity was changed and an all-fiber active similariton cavity was implemented. In this case, the total length of the SMFs between LSMF2 and the input of the Yb3+-doped fiber was reduced to 5 m while the lengths of the Yb3+-doped fiber (LYb) and the proceeding SMF (LSMF2) were kept the same as in Section 2 (1.98 m and 0.4 m, respectively). The schematic diagram of the laser is similar to the one shown in [9]. A power splitter was inserted at the end of LSMF2 with 80% output coupling ratio to keep the value of output pulse energy constant with the advantage of having a high value of peak power as shown in Case 3. The bulky wave plates and polarizer were replaced with an inline fiber polarization controller (PC) and an inline fiber polarizer. The SF was inserted at two positions. The first position is at 0.87 m from the power splitter. The pump power for single-pulse operation was found to be 350 mW. The repetition rate of the pulses was measured to be 27.2 MHz. The average power at the output of the power splitter was measured to be 160 mW. Therefore the pulse energy was calculated to be 5.88 nJ. Figure 12(a) shows the SPD of the output of the power splitter. The spectral bandwidth was measured to be 12.15 nm. The AC profile of the de-chirped pulse is plotted in Fig. 12(b). Using the deconvolution factor calculated in the simulation model, the value of the FWHM of the temporal profile of the de-chirped pulse is 183.15 fs.

 figure: Fig. 12

Fig. 12 Experimental results of all-fiber similariton laser; (a) the mode-locking SPD and (b) de-chirped pulse at the power splitter output.

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The second position of the SF is at 1.34 m from the input of the Yb3+-doped fiber. The pump power, repetition rate and pulse energy were kept unchanged (350 mW and 27.2 MHz, respectively). Figure 12(a) shows the SPD of the output of the power splitter. The spectral bandwidth was measured to be 18.4 nm. The AC profile of the de-chirped pulse is plotted in Fig. 12(b). The value of the FWHM of the temporal profile of the de-chirped pulse is calculated using the same deconvolution faction to be 133.65 fs.

6. Conclusion

In this paper, it is shown that the location of the lumped SF affects the similariton pulse dynamics and characteristics inside the mode-locked laser cavities. Four laser cavities were numerically simulated and two of them were experimentally validated. The spectral bandwidth of the pulse is increased when the narrow bandwidth SF is located close to the Yb3+-doped fiber, but at the expense of the pulse energy. The peak power of the de-chirped pulse depends on the pulse energy, de-chirped pulse width and the degree of the oscillatory structures in the SPD. The combination of these factors results in Case 3 having the highest value of peak power. The experimental results performed on a cavity having the same dimension as in the simulation model supports the simulation results. To ensure that the variation in the spectral bandwidth and the de-chirped pulse width was indeed due to the variation in the SF location, an all-fiber laser cavity was implemented with a different cavity length. The results from the all-fiber active similariton cavity confirm the dependence of the spectral bandwidth and the de-chirped pulse width on the SF location in the cavity irrespective of the total length of the cavity.

References and links

1. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000). [CrossRef]   [PubMed]  

2. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002). [CrossRef]  

3. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004). [CrossRef]   [PubMed]  

4. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010). [CrossRef]   [PubMed]  

5. W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012). [CrossRef]   [PubMed]  

6. S. Lefrancois, C.-H. Liu, M. L. Stock, T. S. Sosnowski, A. Galvanauskas, and F. W. Wise, “High-energy similariton fiber laser using chirally coupled core fiber,” Opt. Lett. 38(1), 43–45 (2013). [CrossRef]   [PubMed]  

7. A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14(21), 10095–10100 (2006). [CrossRef]   [PubMed]  

8. A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber laser,” J. Opt. Soc. Am. B 25(2), 140–148 (2008). [CrossRef]  

9. H. E. Kotb, M. A. Abdelalim, and H. Anis, “An efficient semi-vectorial model for all-fiber mode-locked femtosecond lasers based on nonlinear polarization rotation,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1100809 (2014).

10. J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046605 (2006). [CrossRef]   [PubMed]  

11. E. Ding, E. Shlizerman, and J. N. Kutz, “Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. 47(5), 705–714 (2011). [CrossRef]   [PubMed]  

12. A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71(5), 053809 (2005). [CrossRef]  

13. A. Ruehl, O. Prochnow, D. Wandt, D. Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, “Dynamics of parabolic pulses in an ultrafast fiber laser,” Opt. Lett. 31(18), 2734–2736 (2006). [CrossRef]   [PubMed]  

14. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008). [CrossRef]  

15. X. Li, Y. Wang, W. Zhao, X. Liu, Y. Wang, Y. H. Tsang, W. Zhang, X. Hu, Z. Yang, C. Gao, C. Li, and D. Shen, “All-Fiber Dissipative Solitons Evolution in a Compact Passively Yb-Doped Mode-Locked Fiber Laser,” J. Lightwave Technol. 30(15), 2502–2507 (2012). [CrossRef]  

16. X. Liu, H. Wang, Z. Yan, Y. Wang, W. Zhao, W. Zhang, L. Zhang, Z. Yang, X. Hu, X. Li, D. Shen, C. Li, and G. Chen, “All-fiber normal-dispersion single-polarization passively mode-locked laser based on a 45°-tilted fiber grating,” Opt. Express 20(17), 19000–19005 (2012). [CrossRef]   [PubMed]  

17. H. E. Kotb, M. A. Abdelalim, and H. Anis, “Effect of mode locking technique on the filtering bandwidth limitation in all normal dispersion femtosecond fiber laser,” Proc. SPIE 8961, 89613A (2014). [CrossRef]  

18. H. E. Kotb, M. A. Abdelalim, and H. Anis, “Generalized analytical model for dissipative soliton in all normal dispersion mode locked fiber laser,” IEEE J. Sel. Top. Quantum Electron. (in press).

19. L. Zhang, Z. Zhuo, Z. Pan, Y. Wang, J. Zhao, and J. Wang, “Investigation of pulse splitting behavior in a dissipative soliton fibre laser,” Laser Phys. Lett. 10(10), 105104 (2013).

20. X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A. 81(2), 023811 (2010).

21. X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

22. K. Kieu and F. W. Wise, “All-fiber normal-dispersion femtosecond laser,” Opt. Express 16(15), 11453–11458 (2008). [CrossRef]   [PubMed]  

23. X. Liu and Y. Cui, “Flexible pulse-controlled fiber laser,” Sci. Rep. 5, 9399 (2015).

24. A. M. Weiner, Ultrafast Optics (John Wiley & Sons, Inc., 2009), Chap. 4.

25. Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009). [CrossRef]  

References

  • View by:

  1. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000).
    [Crossref] [PubMed]
  2. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002).
    [Crossref]
  3. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
    [Crossref] [PubMed]
  4. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
    [Crossref] [PubMed]
  5. W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
    [Crossref] [PubMed]
  6. S. Lefrancois, C.-H. Liu, M. L. Stock, T. S. Sosnowski, A. Galvanauskas, and F. W. Wise, “High-energy similariton fiber laser using chirally coupled core fiber,” Opt. Lett. 38(1), 43–45 (2013).
    [Crossref] [PubMed]
  7. A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14(21), 10095–10100 (2006).
    [Crossref] [PubMed]
  8. A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber laser,” J. Opt. Soc. Am. B 25(2), 140–148 (2008).
    [Crossref]
  9. H. E. Kotb, M. A. Abdelalim, and H. Anis, “An efficient semi-vectorial model for all-fiber mode-locked femtosecond lasers based on nonlinear polarization rotation,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1100809 (2014).
  10. J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046605 (2006).
    [Crossref] [PubMed]
  11. E. Ding, E. Shlizerman, and J. N. Kutz, “Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. 47(5), 705–714 (2011).
    [Crossref] [PubMed]
  12. A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71(5), 053809 (2005).
    [Crossref]
  13. A. Ruehl, O. Prochnow, D. Wandt, D. Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, “Dynamics of parabolic pulses in an ultrafast fiber laser,” Opt. Lett. 31(18), 2734–2736 (2006).
    [Crossref] [PubMed]
  14. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
    [Crossref]
  15. X. Li, Y. Wang, W. Zhao, X. Liu, Y. Wang, Y. H. Tsang, W. Zhang, X. Hu, Z. Yang, C. Gao, C. Li, and D. Shen, “All-Fiber Dissipative Solitons Evolution in a Compact Passively Yb-Doped Mode-Locked Fiber Laser,” J. Lightwave Technol. 30(15), 2502–2507 (2012).
    [Crossref]
  16. X. Liu, H. Wang, Z. Yan, Y. Wang, W. Zhao, W. Zhang, L. Zhang, Z. Yang, X. Hu, X. Li, D. Shen, C. Li, and G. Chen, “All-fiber normal-dispersion single-polarization passively mode-locked laser based on a 45°-tilted fiber grating,” Opt. Express 20(17), 19000–19005 (2012).
    [Crossref] [PubMed]
  17. H. E. Kotb, M. A. Abdelalim, and H. Anis, “Effect of mode locking technique on the filtering bandwidth limitation in all normal dispersion femtosecond fiber laser,” Proc. SPIE 8961, 89613A (2014).
    [Crossref]
  18. H. E. Kotb, M. A. Abdelalim, and H. Anis, “Generalized analytical model for dissipative soliton in all normal dispersion mode locked fiber laser,” IEEE J. Sel. Top. Quantum Electron. (in press).
  19. L. Zhang, Z. Zhuo, Z. Pan, Y. Wang, J. Zhao, and J. Wang, “Investigation of pulse splitting behavior in a dissipative soliton fibre laser,” Laser Phys. Lett. 10(10), 105104 (2013).
  20. X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A. 81(2), 023811 (2010).
  21. X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).
  22. K. Kieu and F. W. Wise, “All-fiber normal-dispersion femtosecond laser,” Opt. Express 16(15), 11453–11458 (2008).
    [Crossref] [PubMed]
  23. X. Liu and Y. Cui, “Flexible pulse-controlled fiber laser,” Sci. Rep. 5, 9399 (2015).
  24. A. M. Weiner, Ultrafast Optics (John Wiley & Sons, Inc., 2009), Chap. 4.
  25. Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
    [Crossref]

2015 (1)

X. Liu and Y. Cui, “Flexible pulse-controlled fiber laser,” Sci. Rep. 5, 9399 (2015).

2014 (2)

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Effect of mode locking technique on the filtering bandwidth limitation in all normal dispersion femtosecond fiber laser,” Proc. SPIE 8961, 89613A (2014).
[Crossref]

H. E. Kotb, M. A. Abdelalim, and H. Anis, “An efficient semi-vectorial model for all-fiber mode-locked femtosecond lasers based on nonlinear polarization rotation,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1100809 (2014).

2013 (2)

S. Lefrancois, C.-H. Liu, M. L. Stock, T. S. Sosnowski, A. Galvanauskas, and F. W. Wise, “High-energy similariton fiber laser using chirally coupled core fiber,” Opt. Lett. 38(1), 43–45 (2013).
[Crossref] [PubMed]

L. Zhang, Z. Zhuo, Z. Pan, Y. Wang, J. Zhao, and J. Wang, “Investigation of pulse splitting behavior in a dissipative soliton fibre laser,” Laser Phys. Lett. 10(10), 105104 (2013).

2012 (3)

2011 (1)

E. Ding, E. Shlizerman, and J. N. Kutz, “Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. 47(5), 705–714 (2011).
[Crossref] [PubMed]

2010 (2)

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A. 81(2), 023811 (2010).

2009 (1)

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

2008 (3)

2006 (3)

2005 (1)

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71(5), 053809 (2005).
[Crossref]

2004 (1)

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

2002 (1)

2000 (1)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000).
[Crossref] [PubMed]

Abdelalim, M. A.

H. E. Kotb, M. A. Abdelalim, and H. Anis, “An efficient semi-vectorial model for all-fiber mode-locked femtosecond lasers based on nonlinear polarization rotation,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1100809 (2014).

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Effect of mode locking technique on the filtering bandwidth limitation in all normal dispersion femtosecond fiber laser,” Proc. SPIE 8961, 89613A (2014).
[Crossref]

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Generalized analytical model for dissipative soliton in all normal dispersion mode locked fiber laser,” IEEE J. Sel. Top. Quantum Electron. (in press).

Anis, H.

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Effect of mode locking technique on the filtering bandwidth limitation in all normal dispersion femtosecond fiber laser,” Proc. SPIE 8961, 89613A (2014).
[Crossref]

H. E. Kotb, M. A. Abdelalim, and H. Anis, “An efficient semi-vectorial model for all-fiber mode-locked femtosecond lasers based on nonlinear polarization rotation,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1100809 (2014).

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Generalized analytical model for dissipative soliton in all normal dispersion mode locked fiber laser,” IEEE J. Sel. Top. Quantum Electron. (in press).

Buckley, J.

Buckley, J. R.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Burgoyne, B.

Chan, C. C.

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046605 (2006).
[Crossref] [PubMed]

Chen, G.

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber laser,” J. Opt. Soc. Am. B 25(2), 140–148 (2008).
[Crossref]

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14(21), 10095–10100 (2006).
[Crossref] [PubMed]

Clark, W. G.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Cui, Y.

X. Liu and Y. Cui, “Flexible pulse-controlled fiber laser,” Sci. Rep. 5, 9399 (2015).

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

Ding, E.

E. Ding, E. Shlizerman, and J. N. Kutz, “Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. 47(5), 705–714 (2011).
[Crossref] [PubMed]

Dudley, J. M.

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002).
[Crossref]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000).
[Crossref] [PubMed]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000).
[Crossref] [PubMed]

Ferrari, A. C.

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

Galvanauskas, A.

Gao, C.

Godbout, N.

Han, D.

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002).
[Crossref]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000).
[Crossref] [PubMed]

Hasan, T.

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

Hu, X.

Ilday, F. O.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Kieu, K.

Komarov, A.

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71(5), 053809 (2005).
[Crossref]

Kotb, H. E.

H. E. Kotb, M. A. Abdelalim, and H. Anis, “An efficient semi-vectorial model for all-fiber mode-locked femtosecond lasers based on nonlinear polarization rotation,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1100809 (2014).

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Effect of mode locking technique on the filtering bandwidth limitation in all normal dispersion femtosecond fiber laser,” Proc. SPIE 8961, 89613A (2014).
[Crossref]

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Generalized analytical model for dissipative soliton in all normal dispersion mode locked fiber laser,” IEEE J. Sel. Top. Quantum Electron. (in press).

Kracht, D.

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19(3), 461–469 (2002).
[Crossref]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000).
[Crossref] [PubMed]

Kutz, J. N.

E. Ding, E. Shlizerman, and J. N. Kutz, “Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. 47(5), 705–714 (2011).
[Crossref] [PubMed]

Lacroix, S.

Leblond, H.

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71(5), 053809 (2005).
[Crossref]

Lefrancois, S.

Li, C.

Li, X.

Liu, C.-H.

Liu, X.

X. Liu and Y. Cui, “Flexible pulse-controlled fiber laser,” Sci. Rep. 5, 9399 (2015).

X. Li, Y. Wang, W. Zhao, X. Liu, Y. Wang, Y. H. Tsang, W. Zhang, X. Hu, Z. Yang, C. Gao, C. Li, and D. Shen, “All-Fiber Dissipative Solitons Evolution in a Compact Passively Yb-Doped Mode-Locked Fiber Laser,” J. Lightwave Technol. 30(15), 2502–2507 (2012).
[Crossref]

X. Liu, H. Wang, Z. Yan, Y. Wang, W. Zhao, W. Zhang, L. Zhang, Z. Yang, X. Hu, X. Li, D. Shen, C. Li, and G. Chen, “All-fiber normal-dispersion single-polarization passively mode-locked laser based on a 45°-tilted fiber grating,” Opt. Express 20(17), 19000–19005 (2012).
[Crossref] [PubMed]

X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A. 81(2), 023811 (2010).

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

Lu, H.

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

Mao, D.

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

O’Neill, W.

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

Pan, Z.

L. Zhang, Z. Zhuo, Z. Pan, Y. Wang, J. Zhao, and J. Wang, “Investigation of pulse splitting behavior in a dissipative soliton fibre laser,” Laser Phys. Lett. 10(10), 105104 (2013).

Peacock, A. C.

Popa, D.

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

Prochnow, O.

Renninger, W.

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber laser,” J. Opt. Soc. Am. B 25(2), 140–148 (2008).
[Crossref]

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

Rozhin, A. G.

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

Ruehl, A.

Sanchez, F.

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71(5), 053809 (2005).
[Crossref]

Shen, D.

Shlizerman, E.

E. Ding, E. Shlizerman, and J. N. Kutz, “Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. 47(5), 705–714 (2011).
[Crossref] [PubMed]

Sosnowski, T. S.

Stock, M. L.

Sun, Z.

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

Tang, D. Y.

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046605 (2006).
[Crossref] [PubMed]

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000).
[Crossref] [PubMed]

Tsang, Y. H.

Wandt, D.

Wang, F.

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

Wang, H.

Wang, J.

L. Zhang, Z. Zhuo, Z. Pan, Y. Wang, J. Zhao, and J. Wang, “Investigation of pulse splitting behavior in a dissipative soliton fibre laser,” Laser Phys. Lett. 10(10), 105104 (2013).

Wang, Y.

Wise, F.

Wise, F. W.

S. Lefrancois, C.-H. Liu, M. L. Stock, T. S. Sosnowski, A. Galvanauskas, and F. W. Wise, “High-energy similariton fiber laser using chirally coupled core fiber,” Opt. Lett. 38(1), 43–45 (2013).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber laser,” J. Opt. Soc. Am. B 25(2), 140–148 (2008).
[Crossref]

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

K. Kieu and F. W. Wise, “All-fiber normal-dispersion femtosecond laser,” Opt. Express 16(15), 11453–11458 (2008).
[Crossref] [PubMed]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Wu, J.

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046605 (2006).
[Crossref] [PubMed]

Yan, Z.

Yang, Z.

Zeng, C.

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

Zhang, L.

Zhang, W.

Zhao, J.

L. Zhang, Z. Zhuo, Z. Pan, Y. Wang, J. Zhao, and J. Wang, “Investigation of pulse splitting behavior in a dissipative soliton fibre laser,” Laser Phys. Lett. 10(10), 105104 (2013).

Zhao, L. M.

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046605 (2006).
[Crossref] [PubMed]

Zhao, W.

Zhuo, Z.

L. Zhang, Z. Zhuo, Z. Pan, Y. Wang, J. Zhao, and J. Wang, “Investigation of pulse splitting behavior in a dissipative soliton fibre laser,” Laser Phys. Lett. 10(10), 105104 (2013).

Appl. Phys. Lett. (1)

Z. Sun, A. G. Rozhin, F. Wang, T. Hasan, D. Popa, W. O’Neill, and A. C. Ferrari, “A compact, high power, ultrafast laser mode-locked by carbon nanotubes,” Appl. Phys. Lett. 95(25), 253102 (2009).
[Crossref]

IEEE J. Quantum Electron. (1)

E. Ding, E. Shlizerman, and J. N. Kutz, “Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg–Landau equation,” IEEE J. Quantum Electron. 47(5), 705–714 (2011).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (2)

H. E. Kotb, M. A. Abdelalim, and H. Anis, “An efficient semi-vectorial model for all-fiber mode-locked femtosecond lasers based on nonlinear polarization rotation,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1100809 (2014).

W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012).
[Crossref] [PubMed]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (2)

Laser Photonics Rev. (1)

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008).
[Crossref]

Laser Phys. Lett. (1)

L. Zhang, Z. Zhuo, Z. Pan, Y. Wang, J. Zhao, and J. Wang, “Investigation of pulse splitting behavior in a dissipative soliton fibre laser,” Laser Phys. Lett. 10(10), 105104 (2013).

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (2)

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82(2), 021805 (2010).
[Crossref] [PubMed]

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71(5), 053809 (2005).
[Crossref]

Phys. Rev. A. (1)

X. Liu, “Hysteresis phenomena and multipulse formation of a dissipative system in a passively mode-locked fiber laser,” Phys. Rev. A. 81(2), 023811 (2010).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

J. Wu, D. Y. Tang, L. M. Zhao, and C. C. Chan, “Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046605 (2006).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84(26), 6010–6013 (2000).
[Crossref] [PubMed]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Proc. SPIE (1)

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Effect of mode locking technique on the filtering bandwidth limitation in all normal dispersion femtosecond fiber laser,” Proc. SPIE 8961, 89613A (2014).
[Crossref]

Sci. Rep. (1)

X. Liu and Y. Cui, “Flexible pulse-controlled fiber laser,” Sci. Rep. 5, 9399 (2015).

Other (3)

A. M. Weiner, Ultrafast Optics (John Wiley & Sons, Inc., 2009), Chap. 4.

X. Liu, D. Han, Z. Sun, C. Zeng, H. Lu, D. Mao, Y. Cui, and F. Wang, “Versatile multi-wavelength ultrafast fiber laser mode-locked by carbon nanotubes,” Sci. Rep.3,2718 (2013).

H. E. Kotb, M. A. Abdelalim, and H. Anis, “Generalized analytical model for dissipative soliton in all normal dispersion mode locked fiber laser,” IEEE J. Sel. Top. Quantum Electron. (in press).

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Figures (12)

Fig. 1
Fig. 1 Schematic diagram of the active similariton laser cavity having the SF at four positions to test its effect on the temporal and spectral properties of the pulse; QWP is the quarter wave plate, HWP is the half wave plate and ISO is the Faraday isolator.
Fig. 2
Fig. 2 The numerical normalized temporal profile of the chirped pulse at the end of LSMF2 (SMF2), the through port of the PBS (Pol), the rejection port of the PBS (Output 1) and the coupler output: (a) Case 1, (b) Case 2, (c) Case3, and (d) Case 4. (e) Plot of the FWHM of the chirped pulse
Fig. 3
Fig. 3 The numerical normalized SPD of the chirped pulse at the end of LSMF2 (SMF2), the through port of the PBS (Pol), Output 1 and the coupler output: (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4.
Fig. 4
Fig. 4 (a) Evolution of the total peak power through the four mode-locked laser cavities. (b) Spectral bandwidth at various locations inside the cavity. The locations of LYb, LSMF2, wave plates and polarizer are shown for one of the cases (Case 4) for the sake of clarity. 1: QWP1; 2: HWP; 3: the through port of the PBS; 4: Faraday isolator; 5: QWP2; 6: Lumped SF.
Fig. 5
Fig. 5 (a) The normalized AC of the chirped pulse at Output 1 for Case 2 and Case 3. Temporal profile of the de-chirped pulses: (b) Case 1 and Case 2 and (c) Case 3 and Case 4. Normalized AC of the de-chirped pulses: (d) Case 1 and Case 2 and (e) Case 3 and Case 4.
Fig. 6
Fig. 6 (a) Plot of FWHM of the de-chirped pulse. (b) Plot of peak power of the chirped and de-chirped pulses.
Fig. 7
Fig. 7 Plot of (a) pulse energy at Output 1 and (b) percentage of the pulse energy of the side pulses to the total energy.
Fig. 8
Fig. 8 Evolution of the spectral and temporal RMS widths through the four mode-locked laser cavities: (a) Case 1, (b) Case 2, (c) Case 3 and (d) Case 4. The solid lines represent the u-field, while the dashed lines represent the v-field. 1: QWP1; 2 HWP; 3: the through port of the PBS; 4: Faraday isolator; 5: QWP2; 6: Lumped SF. Virtual lengths are set to these components for the sake of illustration.
Fig. 9
Fig. 9 (a) The numerical normalized temporal profile of the chirped pulse at the through port of the PBS (Pol) and the rejection port of the PBS (Output 1) for Case 2 at Esat equals 6.85 nJ. (b) The corresponding AC profiles at the same value of Esat.
Fig. 10
Fig. 10 (a) Measured normalized transfer function of the lumped SF. (b) Schematic diagram of the diffraction gratings pair combined with a retro-reflector mirror.
Fig. 11
Fig. 11 Experimental mode-locking SPD at Output 1 (blue solid line) and the coupler output (red dashed line) for (a) Case 2 and (b) Case 3. Experimental AC profile of (c) the chirped pulse and (d) de-chirped pulse at Output 1. Case 2: blue solid line and Case 3: red dashed line.
Fig. 12
Fig. 12 Experimental results of all-fiber similariton laser; (a) the mode-locking SPD and (b) de-chirped pulse at the power splitter output.

Tables (2)

Tables Icon

Table 1 Excess Kurtosis factor of the chirped pulses

Tables Icon

Table 2 Chirped pulse energy at the end of LSMF2 and SF loss for each laser cavity

Equations (4)

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u z + i β 2 2 2 u t 2 β 3 6 3 u t 3 + α f 2 u= g( z ) 2 ( 1+ 1 Ω g 2 2 t 2 )u +iγ[ | u | 2 +A | v | 2 ]u+iγB u * v 2 .
v z + i β 2 2 2 v t 2 β 3 6 3 v t 3 + α f 2 v= g( z ) 2 ( 1+ 1 Ω g 2 2 t 2 )v +iγ[ | v | 2 +A | u | 2 ]v+iγB v * u 2 .
g( z )= g o 1+( ( | a | 2 )/ E sat ) .
B W th = 2 2π 1 L[ ( 1ζ ) β 2 a 1 b 1 γ ( l+ a 1 ) Ω g 2 ] .

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