Maximum possible gain

The dashed curve in Fig. 6.3 shows schematically how the local Stokes gain γ(z) is distributed inside a silicon Raman amplifier. As the total amplifier gain is the integral over the dashed curve in Fig. 6.3, the maximum possible total gain is obtained when the

Stokes light “sees” exactly the positive region of the dashed curve between the points marked z₁ and z₂ in Fig. 6.3. Thus, the optimal waveguide length is L_opt = z₂ −z₁ and the optimal pump power is P_opt =P₁, where P₁ is the local pump power at z₁, see Fig. 6.3. We have derived explicit expressions for the optimal length L_opt, the optimal pump powerP_opt and the resulting maximum possible gainG_maxin terms of the material parameters, see Eqs. (6.13), (6.16) and (6.20); they have been published in [RK06].

Derivation

We start by rewriting the amplifier model Eqs. (6.1)–(6.3) in the form 1

P dP

dz = 1 H

dH

dz =−α−B_pH−CH², (6.8)

1 S

dS

dz =−α+B_sH−CH² =γ(z), (6.9) where we have defined the effective local pump intensity

H = P

A_eff, (6.10)

furthermore B_p =β_pp, B_s =g−2β_sp, and

C = ϕλ¯ ²τeffβsp

2hν . (6.11)

In writing Eqs. (6.8)–(6.11), we have assumed that the linear losses for the pump and Stokes waves are equal, αs = αp = α, and we have also neglected the difference in the pump and Stokes wavelengths and have simply set λs = λp = λ. Finally, we have assumed that all three effective areas occuring in the full model (6.1)–(6.3) are equal to Aeff and the confinement factors are unity, see the discussion in Sect. 5.3.1. These approximations often have an error of a few percent only and have the advantage of leading to particularly simple results. Explicit expressions for the general case can be written down, too, but they are lengthy, and the principal behavior remains the same [Ren].

We are interested in the region between z1 and z2, where the local Stokes gain is positive, see Fig. 6.3. Solving the quadratic equation γ(H) = 0, where γ is defined in Eq. (6.9), one finds that the local Stokes gain is positive for local pump intensities betweenH1 and H2, where

H₁ =H(z₁) = B_s+W_s

2C , H₂ =H(z₂) = B_s−W_s

2C , W_s =p

B_s²−4αC. (6.12)

If αC > B_s²/4, W_s is imaginary and the local Stokes gain is never positive, no matter what the local pump power is. In that case, no amplifier can be realized with the given waveguide technology. In the following we assume that αC ≤B_s²/4.

From Eq. (6.12) we immediately get the optimal input pump power required to achieve the maximal possible gain, namely (see Fig. 6.3)

P_opt =H₁A_eff = Bs+Ws

2C A_eff. (6.13)

The optimal length of the waveguide extends from z₁ toz₂, see Fig. 6.3:

Lopt =z2−z1 = Z z2

z1

dz. (6.14)

The integral in Eq. (6.14) can not be solved directly, because we do not knowz₂, which is the location where the pump power has decayed from its initial valueH₁ to the desired value H₂; the decay of the pump power is governed by Eq. (6.8), which has no simple explicit solution. However, we can substitute H(z) for z as the integration variable in Eq. (6.14). Then, using Eq. (6.8), one obtains

L_opt = Z H2

H1

1

H(−α−B_pH−CH²)dH. (6.15) This integral can be solved in closed form as

L_opt = 1

When the waveguide has the optimal length L_opt given by Eq. (6.16) and is pumped with the optimal pump powerP_opt given by Eq. (6.13), we obtain the maximum possible gain,

Again, this integral can not be solved directly, because the longitudinal evolution of the pump intensityH(z) is not known. However, substitutingH(z) for z as the integration variable in Eq. (6.18) and inserting Eqs. (6.8) and (6.9), we obtain

G_max= exp

This integral can now be evaluated explicitly, giving the maximum possible total gain G_max=

a[dB/cm]

Figure 6.4.: (a) Maximal gain G_max, (b) optimal length L_opt and (c) required input pump intensityP_in,opt/A_eff for an optimal non-tapered Raman amplifier versus linear waveguide losses α and effective carrier lifetimeτeff. Remaining parameters: g= 20 cm/GW, β = 0.7 cm/GW, λ= 1550 nm, ¯ϕ= 6×10⁻¹⁰.

Discussion

The maximum possible gain of a silicon Raman amplifier, see Eq. (6.20), depends only onBs, Bp and the product αC, whereC is proportional to the effective carrier lifetime τeff. Both the effective carrier lifetime and the linear waveguide lossesαdepend strongly on how a waveguide is manufactured (see Table 5.3), whereas the remaining parameters contributing to Bs, Bp and C are material properties of bulk crystalline silicon, which are less accessible to deliberate modification. It is therefore interesting to consider the performance of silicon Raman amplifiers as a function of α and τ_eff.

Fig. 6.4a shows the maximum possible gain G_max as a function of the linear losses α and the effective carrier lifetimeτ_eff. According to Eq. (6.20), the maximum possible gain depends only on the product of the two,ατ_eff, so that the curves of constant maximum possible gain in Fig. 6.4a are hyperbola. For ατ_eff ∼ αC > B_s²/4, no amplifier can be realized, which is indicated as the shaded-gray area in Fig. 6.4a.

The fact that the maximum possible gain is determined only by the product ατ_eff is relevant for deciding by which technological steps the performance of a silicon Raman amplifier is to be improved. Our result shows that the maximum possible gain can be increased equally well by decreasing the linear losses α (for example by

waveguide-carrier lifetimeτ_eff by the same factor (for example, by using a p-i-n structure [RJL⁺05], or by Helium implantation [LT06]). On the one hand, when decreasing the linear losses α, the maximum possible gain increases because the pump power can penetrate deeper into the waveguide before it gets so low that the linear losses dominate the Raman gain.

The overall amplifier may thus be longer, and the total gain higher, see Fig. 6.4b. On the other hand, when decreasing the effective lifetime τ_eff, higher pump powers can be tolerated before FCA becomes significant, thus the Raman gain is more dominant which also increases the maximum possible total gain, see Fig. 6.4c.

Finally, the effective area A_eff has no influence on the maximum possible gain or the optimal length. It only scales the pump power necessary to achieve the maximum possible gain, see Eq. (6.13).