Threshold Characteristics Analysis of Dual-end-pumped Nd 3 +-doped Gain-guided and Index-antiguided Fiber Lasers

High power ber lasers have many important applications in industry, defense and commercial fields. However, the major limitations to high power lasers in conventional single mode bers are undesired optical nonlinear effects . The most direct method to solve these problems is increasing the mode area to reduce the optical power density , thus large mode area single-mode fibers have recently attracted a lot of attentions, such as photonic crystal fibers , chirally coupled core fibers [7-8] and so on. However, the maximal obtainable mode field diameter (MFD) is still less than 100 m.In 2003,A.E. Siegman proposed a new type of gain-guided and index-antiguided (GG-IAG) fiber with large mode area, and the maximal MFD reported is about 400 m. In this paper, we report an analytical model for a dual-end-pumped Nd-doped GG-IAG fiber laser, andan exact analytical expression of the threshold pump power for this ber laser has been obtained by solving the improved rate equations (REs) with the additional leakage losses of the signal light. The effects of each parameters of the fiber on the threshold pump power are also discussed.


Introduction
High power ¿ber lasers have many important applications in industry, defense and commercial fields.However, the major limitations to high power lasers in conventional single mode ¿bers are undesired optical nonlinear effects [1][2][3][4] .The most direct method to solve these problems is increasing the mode area to reduce the optical power density [5] , thus large mode area single-mode fibers have recently attracted a lot of attentions, such as photonic crystal fibers [6] , chirally coupled core fibers [7][8] and so on.
However, the maximal obtainable mode field diameter (MFD) is still less than 100ȝm [3] .In 2003,A.E.Siegman proposed a new type of gain-guided and index-antiguided (GG-IAG) fiber with large mode area [9][10] , and the maximal MFD reported is about 400 ȝm [1] .In this paper, we report an analytical model for a dual-end-pumped Nd 3+ -doped GG-IAG fiber laser, and 1 an exact analytical expression of the threshold pump power for this ¿ber laser has been obtained by solving the improved rate equations (REs) with the additional leakage losses of the signal light.The effects of each parameters of the fiber on the threshold pump power are also discussed.

Theoretical analyses
The structure of the GG-IAG fiber is showed in fig. 1.
The refractive index of the fiber cladding n 2 = n 0 , the refractive index difference of the core and the cladding is ǻn (ǻn<0), the power gain factor of the signal light is g.There are leakage losses of signal light and pump light, then according to the literature [11] , a set of improved REs. of Nd 3+ doped dual-end-pumped GG-IAG fiber laser can be expressed as formulas (1), (2)

Results and discussions
All the parameters used in the calculations are given in table1.

The pump light propagation characteristics
The forward and backward pump light can be expressed  Supposing that the maximal (minimal) pump power along the whole fiber is P pmax P pmin , respectively.The allowed fractional variation in the pump light distribution in the fiber core should be below 61% [4] for GG-IAG fiber to maintain single mode oscillation, and which is called uniformity of pump light distribution here and can be defined as: P u =(P pmax -P pmin )/P pmax 100% [4] .P u changes process with the fiber length L is shown in Fig. 4.P u increases with the increasing of L.
When u P is 61%, the allowed maximal fiber length L is about 9.3 cm, which is shown in Fig. 4.
Fig. 4 Pu changes with the fiber length

Threshold pump power characteristics
Supposing the threshold pump power is th P , which changes with N is shown in Fig. 5 and Fig. 6.It can be found that in a certain concentration range, the threshold pump power is lower and keeps nearly invariable, but when the concentration is beyond this range, the threshold increases higher.The reason is when N is lower, it is difficult to supply enough gain to overcome the losses, and when N is too high, the pump power is absorbed fast at the two ends of the fiber, so both make the threshold power increase.In this paper, when L=9cm.a=50ȝm and R 2s =0.75, the minimum of th P is about 11.95W, the corresponding concentration N is about 1.8×10 20 cm -3 .
Fig. 5 The relationship between P th and N with different L Fig. 6 The relationship between P th and N with different a th P changes process with L is shown in Fig. 7. Fig. 7 shows that the threshold pump power is lowest with the optimum fiber length, when the fiber length L is too long or too short, the th P will increases correspondingly.
Because when the fiber is short, the gain is too small to satisfy the oscillation conditions, so the threshold becomes higher.When the fiber is long, the loss of the fiber becomes big, so the threshold becomes higher.
Fig. 7 The relationship between P th and L with different R 2s th P changes process with fiber core radius a is shown in Fig. 8.The optimum fiber core radius a is about 50-55ȝm with different R 2s when L is 9 cm.
Fig. 8 The relationship between P th and a

Conclusions
A dual-end-pumped Nd 3+ -doped GG-IAG fiber laser is analyzed in this work.Pump light propagation and threshold characteristics are both researched.Under the requirement of the uniformity of pump power along the fiber [4] , the maximal fiber length is 9.3cm.The relationship between threshold pump power P th and doped concentration N, fiber length L, fiber radius a is researched respectively.
Simulation results show that there are optimum value of N, a and L where P th is the lowest.When a=50ȝm, L=9cm, N=1.8*10 20 cm -3 , R 1s =1, R 2s =0.75, the lowest threshold pump power P th of GG-IAG fiber laser is about 11.95W, the results of the work may be helpful for the later experiments.

Fig. 1
Fig.1 The schematic diagram of GG-IAG fiber The GG-IAG fiber laser is as shown in Fig.2.Dual-end-pumped model means that pump light (0) p P

Where, N 2 QD
(z) is the upper lasing level population density, N is the concentration of dopant Nd 3+ .ī p and ī s are the pump and signal light frequencies, respectively.A is the cross-sectional area of the core.W is the spontaneous lifetime of the upper-level atoms and the parameter h is Planck's constant.The parameter c is the speed of light in vacuum.represent the fundamental material absorption coefficients of the pump and signal light respectively, p l and s l represent the additional leakage losses of pump and signal light respectively, and they Eqs.(11) and (12a-b), after Eq. (10) is integrated, the threshold pump power can be derived as following: pump light distribution in the fiber core is shown in fig.3.when fiber length L is 9cm, and at the middle of the fiber the total pump power is 2.835W.

Fig. 3
Fig.3 P p distribution along the fiber length )