In order to analytically solve P_{p}(Z), we employ this equation and a following equation,

where η_{q} is a pumping quantum efficiency, ~1 for Yb^{3+}. The two equations give a following equation,

here P_{p}(0) is excitation power at the input edge, and P_{psat} is a pumping saturation power.

P_{p}(z) is a calculated from Eq. 2. In general, Eq. 2 must be numerically solved for the calculation of absorption pumping power by optical fiber length. Provided P_{p}(0) << P_{psat}, Eq. 2 is approximated as follows.

Single pass gain for optical fiber length is expressed by a following equation.

where α_{L} is a damping constant for laser signal light.

The expression of small-signal gain is simplified by Eq. 1. P_{pabs}, a pumping power, that is absorbed by Yb fiber, is obtained from Eq. 1.

In summary, the small-signal gain, G, is obtained as follows.

The second, and third term explains a reabsorption loss, and a propagation loss of fiber core. When the small-signal gain, G, is converted to dB expression, fiber gain is obtained.

### Clad pumping Yb fiber amplifier

We consider a clad pumping Yb fiber amplifier. Provided the core diameter =10 µm, h = 6.63 x 10-34 Js, τ_{f} = 1ms, λ_{p} = 975 nm, λ_{L} = 1064 nm, σ_{e}(λ_{L}) ~2.82×10^{-25}m^{2}, f_{1} = 0, f_{2} = 1, σ_{e}(λ_{L})~0, and the propagation loss α_{L} ~ 0 (under the assumption of four states in long wavelength range), the gain characteristics against absorption pumping power is obtained as shown in Fig. 1. The practical small-signal gain is 30~40 dB owing to a parasitic oscillation and and ASE in the optical fiber.

Fig. 1 Dependency of gain on pumping power of clad-pumped Yb fiber amplifier

### Core direct pumping Yb fiber amplifier

We consider a core pumping Yb fiber amplifier. Provided the core diameter = 6 µm, τ_{f} = 1 ms, η_{q} = 1, λ_{p} = 975 nm, N = 9.5×10^{25}m^{-3}, l = 1 m, and σ_{L} ~ 0, the gain characteristics against signal wavelength, G'(λ_{L}), is obtained as shown in Fig. 2. The pumping absorption power, P_{pabs}, is changed between 0 and 70 mW. For the larger pumping power, the peak wavelength of output light is red-shifted, because the practical small-signal gain is 30~40 dB.

Fig. 2 Core direct pumping Yb fiber amplifier