## Abstract

With the combining effects of the fiber birefringence induced round-trip phase variation and the gain profile reshaping induced spectral filtering in the Erbium-doped fiber laser (EDFL) cavity, the mechanism corresponding to the central wavelength tunability of the EDFL passively mode-locked by nonlinear polarization rotation is explored. Bending the intracavity fiber induces the refractive index difference between orthogonal axes, which enables the dual-band central wavelength shift of 2.9 nm at 1570 nm region and up to 10.2 nm at 1600 nm region. The difference between the wavelength shifts at two bands is attributed to the gain dispersion decided by the gain spectral curvature of the EDFA, and the spacing between two switchable bands is provided by the birefringence induced variation on phase delay which causes transmittance variation. In addition, the central wavelength shift can also be controlled by varying the pumping geometry. At 1570 nm regime, an offset of up to 5.9 nm between the central wavelengths obtained under solely forward or backward pumping condition is observed, whereas the bidirectional pumping scheme effectively compensates the gain spectral reshaping effects to minimize the central wavelength shift. In contrast, the wavelength offset shrinks to only 1.1 nm when mode-locking at 1600 nm under single-sided pumping, as the gain profile strongly depends on the spatial distribution of the excited erbium ions under different pumping schemes. Except the birefringence variation and the gain spectral filtering phenomena, the gain-saturation mechanism induced refractive index change and its influence to the dual-band central wavelength tunability are also observed and analyzed.

© 2014 Optical Society of America

## 1. Introduction

To fulfill the demands of compact and inexpensive ultrafast lasers, passively mode-locked erbium-doped fiber lasers (EDFLs) based on nonlinear polarization rotation mode-locking (NPRML) mechanism have been developed as early as 1990s [1, 2]. The NPRML is induced by the coherently constructive interference of the eigen modes in the fiber, which relies on the self-phase modulation (SPM) effect originated from the intracavity nonlinear refraction index [3]. In contrast to the passive mode-locking by using the real saturable absorbers such as carbon nanotube [4, 5], graphene [6–8], graphite and charcoal nano-particles [9–11], the NPRML is independent of the carrier excitation population of the saturable absorber materials and thus its saturable absorption response is extremely fast. Furthermore, since the SPM effect is broadband, the ultrafast pulse generation is possible over a wide range of wavelengths. Therefore, it permits that the central wavelength of the soliton pulses generated by the NPRML-EDFL can be continuously detuned. Technically, the wavelength shift was achieved by adjusting the polarization controllers to detune the phase difference between two orthogonally polarized components with the help of the polarization-dependent isolator [12]. Although the relationship between the phase delay and the offset central wavelength was theoretically simulated, it was hard to quantitatively define the relationship between the rotation angle of the polarization controllers for the polarization adjustment and the offset central wavelength. Therefore, the quantitative analysis on the central wavelength shifting range is still required to be developed.

In this work, a dual-band central wavelength shift of the NPRML-EDFL system is demonstrated by manipulating the birefringence variations and the gain spectral profile. A precisely defined central wavelength shift can be obtained by accurately adjusting the radius of an intracavity bending single-mode fiber (SMF) circle. The bending SMF circle induces the extra refractive index difference between the fast and slow axes of the SMF. According to the linear transmittance model and the simulation results, the dynamics of wavelength shift is attributed to the cavity filtering originated from the combined effects of the fiber birefringence and the polarization-dependent loss in the EDFL. In the past, some research reports stated that the passively mode-locked EDFLs could be achieved by incorporating some novel materials, and showed the large tunable range of the central wavelength by varying the orientation of an intracavity polarization controller [13–15]. It implies that there are some strong polarization-dependent loss components in the EDFL cavity, and the passive mode-locking mechanism of these research works is probably dominated by the NPRML. Aside from the cavity filtering effect in the EDFL, it is also explored that the gain spectral shape alteration of the EDFL is another mechanism correlated with the central wavelength shift. For a given length of the gain medium, the shape of the gain spectrum can be influenced by adjusting the spatial distribution of the population inversion level of the erbium ions through the EDF. The efficiently accessible method to detune the spatial distribution of the population inversion along the direction of the propagation is varying the direction of the pumping source or varying the amount of the pump power. This operation urges the passively mode-locked EDFL to generate the pulses at the wavelength where the total gain balanced by the total cavity loss. Eventually, it reveals that the wavelength of the NPRML-EDFL can be controlled not only by the intracavity filter caused by the combined effects of the intracavity birefringence and the polarization-dependent loss, but also by the gain spectral profile variation.

## 2. Experimental setup

Figure 1 shows the configuration of NPRML-EDFL system. The total length of single-mode fiber (SMF, Corning, SMF28) used in the EDFL cavity was 4.7 m with the group velocity dispersion (GVD) of β_{2,SMF} = −20 ps^{2}/km. The gain medium was a 2-m long Erbium-doped fiber (EDF, Liekki, Er80-8/125) with the GVD of β_{2,EDF} = −20 ps^{2}/km. Two high-power laser diodes with central wavelength of 980 nm were utilized as the forward and backward pumping source. The 980-nm/L-band wavelength division multiplexing (WDM) couplers were employed for directionally coupling the pumping laser into the EDF from both sides.

The intracavity propagation direction was determined by an isolator which eliminates the backward power. A 1 × 2 optical coupler was utilized for 95% feedback and 5% output. The intracavity polarization was controlled by a polarization controller. After the NPRML was induced via the polarization control, the optical spectra and the autocorrelation traces of the EDFL output were measured by an optical spectrum analyzer (Ando, AQ6317B) and an autocorrelator (Femtochrome, FR-103XL). To increase the refractive index difference between the fast and slow axes of the fiber in the EDFL, a segment of SMF in the EDFL was gradually bended to form a circle with its radius from 3 cm to 1 cm. The experimental configuration of the NPRML-EDFL for studying the influence of the bended intracavity SMF is schematically illustrated in Fig. 1(a). On the other hand, the relationship between the gain spectrum and the central wavelength of passively mode-locked EDFL was also investigated. The experimental configuration of the passively mode-locked EDFL to measure the output under the different pumping conditions (forward, backward and bi-directional) was shown in Fig. 1(b). Moreover, the gain spectral distributions of the EDFL system under the different pumping conditions were measured. Considering that some optical devices such as WDM couplers in the EDFL could possess the wavelength-dependent loss properties, the gain spectrum was measured via the experimental setup demonstrated in Fig. 2. The continuous-wave (CW) light with the power of 1 mW was delivered from the commercial tunable laser (Agilent, 8164A), and the wavelength was tuned from 1550 nm to 1620 nm. Besides, the amplified spontaneous emission (ASE) spectrum was also measured with the same experimental setup.

## 3. Results and discussions

#### 3.1 The gain spectra and the ASE spectra

The gain and the ASE spectra of the EDFA under the different pumping conditions are shown in Fig. 3. Apparently, the peak wavelengths of the gain and the ASE spectra are different under the same pumping conditions. Such a peak wavelength deviation originates from the fact that both spontaneous emission and reabsorption in the insufficiently pumped EDFA occur concurrently to form the ASE spectrum. The ASE is the fluorescence light which is spontaneously emitted from the erbium ions in excited states and subsequently amplified via stimulated emission. Nevertheless, some of the ASE would be reabsorbed because the EDFA is the quasi-three-level fiber amplifiers. Therefore, the ASE spectrum is also correlated with absorption spectrum of the EDFA. Additionally, the peak wavelength of the gain spectrum under the different pumping conditions is significantly deviated at different wavelength. Such a difference results from the non-uniform spatial distribution of the excited-state erbium ions in the EDF, which strictly depends on the direction and the power of the pumping source. Eventually, the gain of the CW light at the different wavelengths is dependent on the power and the direction of the pumping source. The gain spectral profiles corresponding to the fraction of the excited state population in the filterless EDFL has been theoretically investigated in previous work [16].

Furthermore, it should be noted that the gain spectrum of the pulse is different from that of the CW light. When the pulse with a finite bandwidth is amplified in an EDFA with limited gain bandwidth, the gain spectrum of the circulated pulse generally reduces to result in the gain narrowing phenomenon. As a result, the gain narrowing also influences the shortest pulse duration of the pulse. In order to take the gain narrowing effect into account, the master equation which describes the pulse formation dynamics of the passive mode-locking has introduced the term of gain dispersion, *D _{g} = g/*Ω

*, where*

_{g}^{2}*g*is the gain and Ω

*is the bandwidth of the gain line approximated to be parabolic [17]. Specifically, the gain dispersion*

_{g}*D*is the curvature of the gain profile versus frequency at the maximum of the parabolic or Lorentzian shaped gain profile close to the central regime. In addition to the gain bandwidth of the EDF, the magnitude of the gain also decides the strength of the gain-narrowing effect. As the cavity losses which must be balanced by gain are increased in the passively mode-locked EDFL, the effect of the gain narrowing is strengthened. Therefore, the output coupler with small output coupling ratio is utilized in the experimental setup to release such a gain narrowing effect. In conclusion, the gain narrowing effect would affect the parameters of the soliton pulses generated by the EDFL passively mode-locked by the NPR mechanism, including the pulsewidth, the linewidth and the central wavelength.

_{g}#### 3.2 Central wavelength shift based on linear transmittance

With particular adjustment on the polarization controller, the self-started passive mode-locking via the NPR effect could be achieved by enlarging the pump power beyond a threshold of around 95 mW. When the two pump LDs was driven at the maximum pumping current, the total pumping power provided to the EDFL cavity was about 650 mW. In experiment, the central wavelength of the passively mode-locked EDFL is switchable between 1570 nm and 1600 nm, as adjusted by tuning the axis of the polarizer and the state of the polarization controller. The central wavelength shift range is continuously tunable within each band, which is implemented by bending the intracavity SMF circle with its radius gradually varying from 3 cm to 1 cm. Moreover, the wavelength shift becomes significant when shrinking the radius to 2 cm or less, as the birefringence dramatically increases then. Figure 4 demonstrates the optical spectra and the autocorrelation traces of the NPRML-EDFL when the intracavity SMF is bended into a circle with different radii. Obviously, there are two spectral regimes in which the passive mode-locking can be induced in the EDFL with the NPR effect. For comparison, the characteristic parameters of the soliton pulse generated from the passively mode-locked EDFL by bending the intracavity SMF circle at different radii is shown in Table 1. To discuss the effect of birefringence variation induced phase delay and polarization change in more detail, the linear transmittance model is introduced to explain the wavelength tunability of the passively mode-locked EDFL due to the bended intracavity fiber as mentioned above [12].

In principle, the linear transmittance model describes the relative phase delay between the fast (*f*) and slow (*s*) axes of the fiber and determines the transmittance after a round-trip circulation along the EDFL cavity, as given by

*θ*is the angle between the fast axis

_{1}*f*of the fiber and the axis

*P*of the polarizer,

*θ*is the angle between the fast axis

_{2}*f*of the fiber and the axis

*A*of the analyzer, as shown in Fig. 5.

As a result, the net phase delay Δ*ϕ* = Δ*ϕ _{PC}* + Δ

*ϕ*+ Δ

_{LB}*ϕ*is set between the fast

_{NL}*f*and slow axes

*s*of the fiber, Δ

*ϕ*is the phase delay induced by the PC, Δ

_{PC}*ϕ*is the linear phase delay induced by fiber birefringence,

_{LB}*δλ*is the wavelength offset from the central wavelength of

*λ*,

_{s}*L*is the EDFL cavity length,

*L*=

_{b}*λ*/Δ

_{s}*n*is the birefringence beating length, Δ

_{eff}*n*is the refractive index difference between the fast and slow axes of the fiber, Δ

_{eff}*ϕ*is the nonlinear phase delay,

_{NL}*γ*is the fiber nonlinearity, and

*P*is the instantaneous peak power of the NPRML-EDFL pulse. The Eq. (1) clearly signifies that the phase delay between two polarization states is the critical parameter essential for the linear transmittance. In particular, the change of linear transmittance can be introduced by either the polarization controller Δ

*ϕ*, or the fiber birefringence Δ

_{PC}*ϕ*, or the fiber nonlinearity Δ

_{LB}*ϕ*. Among three different phase delay components which affect the net phase delay shown in Eq. (1.a), only the Δ

_{NL}*ϕ*is a function of the wavelength offset (δλ) from the central wavelength. Therefore, the spacing between mode-locked wavelength bands is determined by the length of the EDFL (

_{LB}*L*) and the refractive index difference between the fast and slow axes of the fiber (Δ

*n*) due to the birefringence induced by the bending SMF. The set of Eq. (1) also suggests that the peak linear transmittance is correlated with the Δ

_{eff}*θ = θ*-

_{2}*θ*[12]. Therefore, manipulating the state of polarization controllers as well as the angles of polarizer and analyzer in the EDFL cavity would alternate the linear transmittance accordingly. To simplify the analysis, we assume that the initial polarization state of the circulated pulse in the NPRML-EDFL is

_{1}*θ*=

_{1}*π*/4 and

*θ*=

_{2}*θ*+ Δ

_{1}*θ.*The Fig. 6 shows the transmittance curves of the EDFL versus the total phase delay added between the two polarization components (polarizer and analyzer) with different Δ

*θ*values, as derived from the aforementioned linear transmittance model.

When the Δ*θ* is equivalent to π/2, the maximum linear transmittance occurs at certain phase delay repeated by a period of an integer multiple of 2π radians. Correspondingly, the passive mode-locking would easily occur at wavelengths near 1600 nm where the gain is small, as shown Fig. 3(a). On the other hand, if the Δ*θ* is equivalent to 5π/16, the maximum linear transmittance occurring at certain phase delay with a period of an integer multiple of 2π radians is reduced. In this case, the passive mode-locking would not be induced at 1600 nm since the corresponding gain is insufficiently high to overcome the mode-locking threshold. Instead, the 1570 nm regime becomes another wavelength window with the maximized linear transmittance, where the gain is extremely high so that the mode-locking threshold can be overcome and the passive mode-locking can be induced eventually. To further analyze the relationship between the central wavelength offset and the bended intracavity SMF, we assume that the initial polarization state of the light in the passively mode-locked EDFL is *θ _{1}* =

*π*/4 and

*θ*=

_{2}*θ*+

_{1}*π*/2. If the SMF is circularly bended, the refractive index difference Δ

*n*between the fast and slow axes of the fiber would be increased. In addition, the linear phase delay induced by the fiber birefringence

_{eff}*Δϕ*is also increased, which can be expressed as Δ

_{LB}*ϕ*= Δ

_{LB}’*ϕ*+ Δ

_{LB}*ϕ*As a result, with more

_{b}.*Δϕ*induced by seriously bending the intracavity SMF, the maximal transmittance would be shifted toward shorter wavelength. The transmittance coefficient versus phase delay at different phase delay conditions caused by bending the intracavity SMF into a circle are shown in Fig. 7(a).

_{b}According to the linear phase delay induced by fiber birefringence (Δ*ϕ _{LB}*) expressed in Eq. (1.c), if the phase delay induced by the increased deviation on the refractive index difference (Δ

*n*) between the fast and slow axes of the intracavity SMF is positive, the wavelength offset (

_{eff}*δλ*) from the original wavelength will be negative. Therefore, the mode-locked wavelength would shift toward the shorter wavelength when the intracavity SMF circle is tightly bended. In addition, it is clearly seen that the pulsewidith is broadened when the bending radius is as small as 1 cm, which corresponds with the fact that the bandwidth of the linear transmittance decreases when the birefringence is increased based on the linear transmittance [12]. The reduced bandwidth of the linear transmittance finally limits the spectral linewidth and broadens the pulsewidth of the passively mode-locked EDFL. Although the central wavelength can be further shifted by seriously bending the intracavity SMF, the bending loss would be dramatically increased when the bending radius becomes less than 1 cm [18]. Therefore, such a large bending loss would reduce too much energy of the light circulated in EDFL to overcome the passive mode-locking threshold. Thus, the seriously bending loss confines the tunable range of the central wavelength of NPRML-EDFL. Apparently, the tunable ranges of the central wavelength are not equivalent at 1570 nm and 1600 nm regimes even with the same variation range on the bending radius, as shown in Figs. 7(b) and 7(c). It is derived from the fact that the gain spectrum at wavelength regime around 1570 nm is nearly uniform, leading to a relatively small change on the gain dispersion. In contrast, the slope of the gain spectrum at wavelength regime around 1600 nm is relatively sharp, hence the change of the gain dispersion would be relatively large, and the central wavelength requires a shift toward the shorter wavelength where the total gain is larger to overcome the passive mode-locking threshold.

#### 3.3 Central wavelength shift based on pumping geometry

In principle, the effects of gain dispersion and gain saturation on the pulse propagating through the EDFA can be theoretically elucidated as below [19]. Basically, the equation used to described the propagated pulse in the EDFA can be written as

*v*denotes the group velocity,

_{g}*β*

_{2}the group velocity dispersion of the EDF,

*g*the EDFA gain,

*α*

_{int}the cavity loss, and α the linewidth enhancement factor. Typically, the gain-induced refractive index change is dominated by the parameter α. In general, the time-dependent dispersion and saturation phenomena on the optical gain are involved in the term of

*g*in Eq. (2). The gain dispersion effect originates from the pulse with its spectral linewidth close to the gain bandwidth. This leads to the gain narrowing effect as the components far away from central frequency ω

_{0}fail to be equivalently amplified. Assuming that the peak frequency ω

_{0}of the pulse is coincident with that of the gain, i.e.

*g*(ω

_{0})≡

*g*

_{p}, the first-order derivative term at gain peak equals to zero

*g’*= 0 and the second-order derivative term at gain peak is negative

*g”*<0. The

*g*(ω) can be expanded by Taylor series and approximated as

After transforming the g(ω) back to time domain by replacing the ω-ω_{0} with $i(\partial /\partial t)$,

_{0}is the small-signal gain at unsaturated condition, τ

_{c}the carrier lifetime and E

_{sat}= η

*ω*

_{0}

*σ*/

*a*the saturation energy of the EDFA with σ denoting the cross-section and

*a*the gain coefficient of the central mode. Deriving the approximated solution of Eq. (5) leads to

Figure 8 demonstrates the optical spectra and the autocorrelation traces of the NPRML-EDFL under different pumping geometries. When the passive mode-locking occurs in the wavelength regime around 1570 nm under dual-directional pumping condition, the EDFL delivers the soliton pulses with the pulsewidth of 370 fs and the corresponding spectral linewidth of 7.26 nm. In contrast, when the passive mode-locking is induced at about 1600 nm under dual-directional pumping condition, the pulsewidth is 360 fs; moreover, the spectral linewidth slightly broadens to 7.51 nm. All the TBPs under different pump geometries and at different wavelength regions are larger than the transformed-limited value of 0.315, indicating that the soliton pulses are slightly chirped. The parameters of the soliton pulse generated by the NPRML-EDFL under different pumping geometries are summarized in Table 2.

It is noted that the pulsewidth is shortest when the passively mode-locked EDFL is under dual-directional pumping condition. This result can be attributed to the fact that when the dual-directional pumping geometry is introduced, fewer erbium ions in the ground-state can reabsorb the soliton pulses in the EDFL cavity, inducing fewer loss in the nearly completely pumped EDF. Therefore, the reduced cavity loss would lessen the gain narrowing effect, and hence the pulsewidth of the soliton pulses would be shortened. As evidence, it is found that the central wavelength of the passively mode-locked EDFL can be detuned by alternating the pumping geometries of EDF. If the forward pumping scheme for the EDFL is transferred to the dual-directional pumping condition with the passive mode-locking wavelength set at 1600 nm region, the central wavelength of the mode-locked spectrum shows a red-shift from 1601.5 nm to 1604.5 nm. Similarly, when the backward pumping scheme is replaced by the dual-directional pumping scheme, the central wavelength also red-shifts from 1602.6 nm to 1604.5 nm. These observations can be principally interpreted by the equation derived in Eq. (7). In general case, the last gain saturation term on the right-hand side of Eq. (7) cannot be negligible when the input pulse energy arises close to the saturation energy of the EDFA, which eventually induces a red-shifted output spectrum according numerical derivation [19]. This spectral red-shift phenomenon is attributed to the SPM under the gain-saturation induced refractive index change [20], which has also been observed in the passively mode-locked EDFL system [21].

As opposed to previous works, it is also discovered that a blue-shift phenomenon would occur when increasing the gain of the EDFL. When the passive mode-locking was set at about 1570 nm, a similar spectral red-shift from 1570.9 nm to 1573.3 nm was observed after transferring the pumping scheme from the forward to the dual-directional case. On the contrary, the spectral peak exhibits a special blue-shift when the backward pumping was changed to the dual-directional pumping scheme, providing a spectral shift from 1576.8 nm to 1573.3 nm. This particular blue-shift phenomenon originates from the fact that the gain dispersion effect becomes the dominating factor in this case. The gain spectrum and the central wavelength of the passively mode-locked EDFL under different pumping geometry conditions are shown in Fig. 9(a). It is clearly seen that the central wavelength of the mode-locked pulse is not exactly located at the wavelength with maximum gain. Besides, the curvature of the gain profile where the passive mode-locking occurs is altered when the different pumping geometries is chosen, indicating the effect of the gain dispersion can be detuned by changing the pumping geometries. Since the central wavelength is located at the wavelength where the total gain *g _{total}* =

*g*(1 + 1/Ω

_{g}

^{2}) most easily match the total loss. Therefore, the central wavelength will be determined by the effect of gain dispersion when the variation of the curvature of the gain profile is distinct. Moreover, the effect of the gain dispersion also leads to the fact that even the forward and backward pumping geometries supply the same pump power, the NPRML-EDFL still shows the different central wavelength due to the different gain profile. It is concluded that besides the gain saturation, the gain dispersion accompanied with the wavelength-dependent transmittance are the decisive factors to determine the central wavelength of the NPRML-EDFL.

## 4. Conclusion

The wavelength-tunable NPRML-EDFL setup is demonstrated by simply bending the intracavity SMF into a circle with the shrinking radius from 3 cm to 1 cm to enlarge the fiber birefringence. The tunable blue-shift range of the soliton pulse is 2.9 nm at 1570 nm region and 10.2 nm at 1600 nm region. According to the linear transmittance model analysis on cavity loss, the tunability of the central wavelength is attributed to the combining effects of fiber birefringence and cavity filtering in the NPRML-EDFL. Besides, the rotation angle of fast (slow)-axis polarization due to the adjustment on the PC could change the angle between fast (or slow) and polarizer axes after a round-trip, leading to the cavity filtering loss varied. Hence, the variation of the filtering loss accompanying with the wavelength-dependent gain profile would provide a spectral hopping of the soliton pulse at dual bands between 1570 nm and 1600 nm. Moreover, such a large difference between the central wavelength shifting ranges obtained at aforementioned two bands reveals the fact that the gain dispersion is significantly different at these bands. The gain dispersion phenomenon originates from the gain narrowing effect caused by ultrashort pulse amplification in the EDFA with the limited gain bandwidth. As a result, the gain narrowing effect depends on the shape of the gain profile. On the other hand, by changing the pumping geometry, the variation on the spatial distribution of gain profile effectively introduces a reshaped gain curvature, leading to another wavelength shift mechanism. The alteration on pumping geometry from forward to backward condition with the same pump power also results in a central wavelength shift of the soliton pulse as much as 5.9 nm at 1570 nm regime and 1.1 nm at 1600 nm regime. In contrast to the spectral red-shift due to refractive index change induced by gain-saturation, a novel blue-shift phenomenon based on the gain dispersion with the increased pump power is observed. After transferring backward pumping geometry to dual-directional pumping geometry, the central wavelength is blue-shifted from 1576.8 nm to 1573.3 nm. In this case, because the gain profile with lower curvature possess the broader gain bandwidth and the central wavelength of NPRML-EDFL must be located at the wavelength where the total gain most easily match the total loss, the gain reshaping effect inevitably forces the NPRML-EDFL to shift toward the wavelength shorter than that of the original case. In conclusion, by simply manipulating the key parameters such as fiber birefringence and pumping geometry, the tunable central wavelength of the NPRML-EDFL is available. The central wavelength tunability of the NPRML-EDFL is determined by three main effects, i.e. the linear transmittance, the gain saturation and the gain dispersion mechanisms.

## Acknowledgments

This work was financially supported by Ministry of Science and Technology, Taiwan under Excellent Research Projects of National Taiwan University, Taiwan under grants NSC 101-2221-E-002-071-MY3,NSC 102-ET-E-002-008-ET, MOST 103-2221-E-002-042-MY3, and 103R89083.

## References and links

**1. **K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. **28**(24), 2226–2228 (1992). [CrossRef]

**2. **V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, “Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarization rotation,” Electron. Lett. **28**(15), 1391–1393 (1992). [CrossRef]

**3. **H. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. **28**(10), 2086–2096 (1992). [CrossRef]

**4. **S. Y. Set, H. Yaguchi, Y. Tanaka, and M. Jablonski, “Laser mode locking using a saturable absorber incorporating carbon nanotubes,” J. Lightwave Technol. **22**(1), 51–56 (2004). [CrossRef]

**5. **F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. **3**(12), 738–742 (2008). [CrossRef] [PubMed]

**6. **Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. **19**(19), 3077–3083 (2009). [CrossRef]

**7. **Z. Han, B. Qiaoliang, T. Dingyuan, L. Zhao, and K. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. **95**(14), 141103 (2009). [CrossRef]

**8. **P. L. Huang, S.-C. Lin, C.-Y. Yeh, H.-H. Kuo, S.-H. Huang, G.-R. Lin, L.-J. Li, C.-Y. Su, and W.-H. Cheng, “Stable mode-locked fiber laser based on CVD fabricated graphene saturable absorber,” Opt. Express **20**(3), 2460–2465 (2012). [CrossRef] [PubMed]

**9. **G.-R. Lin and Y.-C. Lin, “Directly exfoliated and imprinted graphite nano-particle saturable absorber for passive mode-locking erbium-doped fiber laser,” Laser Phys. Lett. **8**(12), 880–886 (2011). [CrossRef]

**10. **Y.-H. Lin and G.-R. Lin, “Free-standing nano-scale graphite saturable absorber for passively mode-locked erbium doped fiber ring laser,” Laser Phys. Lett. **9**(5), 398–404 (2012). [CrossRef]

**11. **Y.-H. Lin, Y.-C. Chi, and G.-R. Lin, “Nanoscale charcoal powder induced saturable absorption and mode-locking of a low-gain erbium-doped fiber-ring laser,” Laser Phys. Lett. **10**(5), 055105 (2013). [CrossRef]

**12. **W. S. Man, H. Y. Tam, M. S. Demokan, P. K. A. Wai, and D. Y. Tang, “Mechanism of intrinsic wavelength tuning and sideband asymmetry in a passively mode-locked soliton fiber ring laser,” J. Opt. Soc. Am. B **17**(1), 28–33 (2000). [CrossRef]

**13. **H. Zhang, D. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. **96**(11), 111112 (2010). [CrossRef]

**14. **D. Popa, Z. Sun, T. Hasan, F. Torrisi, F. Wang, and A. C. Ferrari, “Graphene Q-switched, tunable fiber laser,” Appl. Phys. Lett. **98**(7), 073106 (2011). [CrossRef]

**15. **C. Zhao, Y. Zou, Y. Chen, Z. Wang, S. Lu, H. Zhang, S. Wen, and D. Y. Tang, “Wavelength-tunable picosecond soliton fiber laser with Topological Insulator: Bi_{2}Se_{3} as a mode locker,” Opt. Express **20**(25), 27888–27895 (2012). [CrossRef] [PubMed]

**16. **P. Franco, M. Midrio, A. Tozzato, M. Romagnoli, and F. Fontana, “Characterization and optimization criteria for filterless erbium-doped fiber lasers,” J. Opt. Soc. Am. B **11**(6), 1090–1097 (1994). [CrossRef]

**17. **H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B **8**(10), 2068–2076 (1991). [CrossRef]

**18. **Q. Wang, G. Rajan, P. Wang, and G. Farrell, “Polarization dependence of bend loss for a standard singlemode fiber,” Opt. Express **15**(8), 4909–4920 (2007). [CrossRef] [PubMed]

**19. **G. P. Agrawal, “Effect of gain dispersion on ultrashort pulse amplification in semiconductor laser amplifiers,” IEEE J. Quantum Electron. **27**(6), 1843–1849 (1991). [CrossRef]

**20. **G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. **25**(11), 2297–2306 (1989). [CrossRef]

**21. **Y.-H. Lin, J.-Y. Lo, W.-H. Tseng, C.-I. Wu, and G.-R. Lin, “Self-amplitude and self-phase modulation of the charcoal mode-locked erbium-doped fiber lasers,” Opt. Express **21**(21), 25184–25196 (2013). [CrossRef] [PubMed]