should be low due to the fiber loss. Therefore, the theoretical results of this paper expect that there is an optimum fiber length (at the middle fiber length 10 m) which provides both maximum slope efficiency S and minimum threshold pump power Pin, as indicated in Figure 2. This conclusion is more reasonable than that by Shimizu and can be used to design optical fiber lasers as well as the optical fiber amplifier with four-level material. 4. CONCLUSION A theoretical model based on the rate equations of the four-level has been set up to analyse the four-level single mode fiber laser. The numerical results are in good agreement with available experimental results by Shimizu [7]. It is found that there is an optimum fiber length which provides the maximum slope efficiency S and the minimum threshold pump power of the fiber laser. This conclusion is more reasonable than that of Shimizu [7] and is applied to the design of the optical fiber laser as well as optical fiber amplifier with four-level material. APPENDIX As is well known, the power density distribution in the single mode fiber with refraction index distribution of n y o)-n y u) ] where p (r , z) and p(0, z) are the power density at (r , z) and (0, z) , respectively. The power in the single mode fiber at z is then or P(2) = [ aTru2/(a: + z) ] p (0 , z) (A-2) where use has been made of (A-1) in deriving (A-2) From (A-2) and (A-1), p (0 , 2) = [ (a + 2) / a r u 2 ] p (.) and Suppose that the dB value of the optical fiber loss per meter is Dp. Then ~ (z) = Pin,(0)exp(-D,Z/IO) and P (r , z) = [ l-(r / u) * ] [ (* + 2)/asu2] xPi,,(0)exp[-D , z / ~ o ] Therefore, the energy density of the fiber at (r , z) is REFERENCES 1. R. J. Mears, L. Reekie, S. B. Poole, and D. N. Payne, "Develop-ment of Rare-Earth-Doped Fibres and Single-Mode Fibre Lasers," + 2 6 2h €3 CHARACTERISTIC IMPEDANCE z; = __-1 i 2% 1-+ 0.441 + 0.082-x (1.451 + In(+ 0.94))] ohm I W' h for-2 1 (2)
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