Behaviour of Non-ideal Saturable Absorber Doped on Active Fiber

In this paper, a non-ideal saturable absorber doped with an active fiber device is considered. The differential equations that describe the behavior of the device are simplified and solved numerically. The results show that as the non-ideal term (loss) increases, the laser performance degradation increases.


Introduction
In the fiber laser with a non-ideal distributed saturable absorber (FLDSA), two different types of particles are distributed through the fiber core. These two types of particles interact via the electric field in the fiber core [1,2]. Different work has been done in FLDSA due to the fact that FLSDA is a simple structure that exhibits laser pulsing behavior, Q-switching, and mode locking in the range of picoseconds [3][4][5][6][7][8][9][10][11]. In addition, the FLSDA structure is compatible with the existing fiber networks.
Passive mode locking relies on the action of the distributed saturable absorber where its material becomes more transparent to light as the light intensity increases. As a result, if a saturable absorber is placed inside the fiber laser cavity, it produces modulated loss at a separation frequency of l c , where l is the fiber length and c is the speed of light.
In the previous work two different approaches have been used to describe the pulsing behavior under the assumption that the saturable absorber is located as a bulk cell near the end of the fiber and the absorber is an ideal one. In the first approach, the electric field, active medium, and passive medium loss were expanded by Taylor expansion about the maximum to determine the expansion coefficients from the self-consistency condition [7][8][9][10]. In the second approach a functional form for the pulse profile was chosen and then its parameters on the basis of a second order approximation was determined [11,12]. In this paper the passive mode locking in the non-ideal saturable absorber is described with a mathematical model for pulsing behavior. In this case, the pulse width is usually many orders shorter than the mode separation. A non-ideal saturable absorber is introduced for pulse evaluation, and distributed uniformly in the fiber core thereby reducing the separation frequency between modes. In this approach the three coupled nonlinear differential equations that describe the field inside the fiber, active medium gain, and passive medium loss, are simplified using the slowly varying field approximation. The resulting system is solved numerically.

Mathematical Model
It is assumed that the active medium is a two level homogeneously broadening system which introduces gain only and distributed uniformly inside the fiber core. The same assumptions apply to non-ideal saturable absorber, except that it introduces attenuation instead of gain with nonideal term. The governing equations that describe the propagation of the time dependent electric field pulse envelope ( ) A t , the active medium gain and the passive medium losses are given by [2]: Where RT T is the recovery time for generated pulses, g is the gain medium parameter, a ω ∆ is the bandwidth of the linear resonant filter, δ is the self-phase modulation and L is the constant fiber losses, including Fresnel losses faces, scattering, and absorption losses within the fiber core (additional losses), and the terms: , describes the saturable absorber medium with self-phase modulation (SPM), and to account for the effects of SPM and GVD, and the effects of absorption with gain, respectively. Due to the fact that SA τ ≻ sol τ , then the pulse shape is not strongly influenced by the saturation absorber speed; slower saturable absorber can be used to obtain short pulses [2]. The saturable absorber balance against gain can be obtained by dropping the imaginary terms that corresponds to SPM and GVD from equation (1): The gain medium is described by [2]: Where O g is the small signal gain or totally unsaturated gain, 1 ( ) g t is the totally saturated gain, I1sat is the saturation intensity of the active medium, and ( ) I t is the light intensity of the output mode locking pulse which can be normalized and written as [2]: The non-ideal saturation absorber can be recognized in the form: Where 0 S is the non-ideal absorption term, and ( ) t σ is the saturable term given by: G s (t) is the pump parameter given in [9] by, Where I p (t) is the pump pulse intensity function which is assumed to be in the form: Where I po is the pump pulse amplitude and t p is the pump pulse duration.
The gain for active medium far from saturation can be represented by: where G s1 is a new constant equal to 0 RT g T . Using the slowly varying field approximation for equation (4), and using equations (6-12) the resulting equations after some manipulations can be written as [1][2][3].
Let ( ) and 0 0 1 Then equation (14) can be rewritten as: Which describes the active fiber with non-ideal absorber. If a s R w G = ∆ , then equation (17) can be rewritten as: Equation (18) can be solved numerically for proper values of media constants, active fiber medium, and non-ideal absorber medium. The nonlinear term in equation (18) involving 3 A plays an important role in determining the spectral properties of the laser field. However, if 3 A is neglected, the spectrum of ( ) A t will correspond to a pumping pulse going through a linear resonant filter with a bandwidth a ω ∆ . This is the case below the threshold where the cubic term can be neglected [14].

Results and Discussions
The behavior of non-ideal saturable absorber doped on active fiber can be described accurately by equation 18. This equation accounts for the effects of passive/active fiber mediums, and the pumping characteristics. Figure 1 shows the typical saturable absorber behavior for ideal and non-ideal absorbers. It is clear that as the irradiance increases, the absorption coefficient decreases. The reduction rate for the non-ideal absorber is less than that for the ideal absorber.    Figure 4 shows the effect of R on pulsing behavior. As expected, as R increases the pulse amplitude is also increases in the pulsing region, and the pulse width decreases. This approach can be used to generate pulses with small duration. The effect of linear dispersion parameter 1 a on the side pulses can be clearly seen in figures 5, 6, 7 and 8. As 1 a decreases, the side pulses increase faster

Conclusion
Using active fiber laser with non-ideal absorber, the behavior of generated pulse is strongly depending on the absorber parameters. As the non-ideal (non-saturable) term increases, the pulse performance degradation increases; pulse amplitude deceases and pulse width increases. The effect of non-saturable parameter and fiber loss on parameter a is the same. This makes the possibility to use high loss fiber with very small non-saturable term or vice versa. By introducing the linear dispersion parameter, the ideal term increases and the effect of non-ideal term is compensated.