Co- and counter-propagating effective areas

We now derive the effective areas for stimulated Raman scattering in silicon waveguides.

Eq. (2.42) describes the longitudinal evolution of the forward- and backward-propagating Stokes powers in the waveguide in a general way, and we will rewrite the SRS contribution to the four nonlinear interactions described by Γ_s+p+, Γs+p−, Γs−p+and Γs−p−as the ratio of the bulk Raman-gain constant and certain effective areas, as illustrated in Sect. 2.2.5.

We consider the case where the pump and Stokes frequency difference is such that the Raman gain is maximal, i. e., ω_p−ω_s=ω₀.

We consider only silicon waveguides oriented along the [011] direction on a (100) surface, as in these the quasi-TE copolarized gain is maximal, see Sect. 5.2.1. Thus, the nonlinear susceptibility tensor of silicon, Eq. (5.1), must first be transformed from the crystallographic-axes coordinate system ˜x,y,˜ z˜to the waveguide coordinate systemx, y, z (the relation between the two coordinate systems is shown in Fig. 5.2, with θ = 45^◦),

yielding

is the change-of-bases transformation matrix. With the help of the intrinsic permutation symmetry of χ^(3),SRS_ijkl to adjust the order of the frequency arguments, Eq. (5.7) can be inserted into Eq. (2.44) to get the modal gain coefficient for stimulated Raman scattering from the forward-propagating pump wave to the forward-propagating Stokes wave,

4 Re Γ_s+p+= 30ωs Following the idea of Sect. 2.2.5, we want to rewrite the modal gain coefficient (5.9) as

4 Re Γ_s+p+ = g_R(ω₀)

A^(SRS)_eff,co , (5.10)

where g_R(ω₀) is the peak bulk gain coefficient, see Eq. (5.6), and A^(SRS)_eff,co is the effective area to be derived. Solving Eq. (5.10) for the latter and inserting Eq. (5.7), one obtains

A^(SRS)_eff,co = 4Z₀²NˆpNˆs where e_p,s are the pump and Stokes mode fields with real transverse and imaginary longitudinal components (see Eqs. (2.11)–(2.12)), and ˆN_p,s are the corresponding mode normalizations defined in Eq. (2.14).

We can proceed similarly for the other SRS contributions occurring in Eq. (2.42) and write them as the ratio of the bulk gain given by Eq. (5.6) and an effective area.

The result is that the other co-propagating SRS contribution 4 Re Γs−p−, describing the gain exerted on the backward-propagating Stokes wave by the backward-propagating pump wave, has the same effective area A^(SRS)_eff,co given in Eq. (5.11). The two counter-propagating SRS contributions 4 Re Γs+p− and 4 Re Γs−p+, however, share an effective

Figure 5.3.: The thick solid and dashed curves show the effective areas A^(SRS)_eff,co and A^(SRS)_eff,cntr for co- and counter-propagating SRS as a function of the pump wavelength λp. The curves correspond to the silicon rib waveguide on the right-hand side of Fig. 5.1. All modes are quasi-TE.

area which differs from Eq. (5.11) in two minus signs:

A^(SRS)_eff,cntr= 4Z₀²Nˆ_pNˆ_s n_pn_s

Z

Si

(e^x_s)²[(e^y_p)²−(e^z_p)²] + (e^y_s)²[(e^x_p)²+ (e^y_p)²] + (e^z_s)²[(e^x_p)²−(e^z_p)²] + 2(e^x_se^y_se^x_pe^y_p−e^x_se^z_se^x_pe^z_p−e^y_se^z_se^y_pe^z_p) dA

⁻¹

. (5.12) In summary, the contribution of SRS to the longitudinal evolution of the forward- and backward-propagating Stokes powers P_s^± can be written

± 1 P_s^±

dP_s^±

dz =g_R(ω₀) P_p^± A^(SRS)_eff,co

+ P_p^∓ A^(SRS)_eff,cntr

!

. (5.13)

The thick solid and dashed curves in Fig. 5.3 show the effective areas for co- and counter-propagating SRS as a function of the pump wavelength. They have been calcu-lated using a custom-made full-vectorial mode solver (see Appendix A) for the quasi-TE fundamental mode of the waveguide used by Intel [RJL⁺05], the geometry of which is shown on the right-hand side of Fig. 5.1 (the waveguide is clad on top with silica, which is not shown in the figure). In calculating the effective areas for the various wavelengths, the Stokes wavelength is always offset from the pump wavelength by the Raman shift of ω₀. Fig. 5.3 shows that the difference between the forward- and backward-propagating effective areas can be as large as 5% at λ_p = 3µm for this waveguide, while becoming smaller at shorter wavelengths.

So far we have only written the SRS “seen” by the Stokes powers in terms of effective

expressions for the terms that appear in the equations for the longitudinal evolution of the forward- and backward-propagating pump powers, see Eq. (2.41). The result can be written in the form

which follows from the symmetry relation

˜

χ^(3),SRS_ijkl (ω_p, ω_s,−ω_s) =h

˜

χ^(3),SRS_ijkl (ω_s, ω_p,−ω_p)i∗

, (5.15)

which can be obtained from the theory of [SB65] that has also led to Eq. (5.1). As in section 2.3.1, the factor λ_s/λ_p >1 occurring in Eq. (5.14) expresses photon-number conservation.

Polarization dependence of the effective areas for SRS

As discussed in section 2.2.5, the effective area is defined such that it encapsulates all the information about the tensorial structure of the nonlinearity, in this case SRS: one simply inserts the pump and Stokes mode fields of the waveguide into the formula. As an example, we have shown in Fig. 5.4 the effective area for co-propagating SRS as a function of the pump wavelength.

There are four curves, each corresponding to a particular combination of pump and Stokes polarizations. The thin solid curve in Fig. 5.4 (also shown in Fig. 5.3 as the thick solid curve) shows the effective area when both the pump and the Stokes powers are guided in the fundamental quasi-TE mode of the waveguide. The other two thin curves in Fig. 5.4 represent the effective areas for the cases when one of the pump and Stokes modes is the quasi-TE mode, and the other one is the quasi-TM mode. The three thin curves in Fig. 5.4 almost coincide and increase slightly towards larger wavelengths due to the decreasing mode confinement.

Finally, the thick solid curve in Fig. 5.4 represents the case where both the pump and the Stokes powers are guided in the fundamental quasi-TM mode of the waveguide.

The effective areas are much larger than for the other polarization combinations. Thus SRS is very inefficient, in accordance with the discussion in section 5.2.1. In contrast to the other curves, however, the thick solid curve in Fig. 5.4 decreases towards larger wavelengths. On the one hand, the mode confinement decreases for larger wavelengths and this should increase the effective area. However, at the same time, the hybridicity of the mode becomes stronger—with increasing wavelength, theyandz components of the electric field of the quasi-TM mode become larger, and these components can contribute

Figure 5.4.: Illustration of the strong polarization dependence of SRS in silicon waveguides:

the figure shows the effective areas A^(SRS)_eff,co for co-propagating SRS as a function of the pump wavelength. TM–TM amplification (thick solid line) is much less efficient than TE–TE, TE–

TM, or TM–TE amplification.

to the amplification of the quasi-TM Stokes mode. As a result, the effective area for TM–TM amplification decreases with increasing wavelength.