Theory of four-wave mixing for bound and leaky modes

We present a comprehensive theory of four-wave mixing in waveguide geometries, providing a rigorous description of the dynamics of both bound as well as leaky modes within a single theoretical framework. Our approach is based on the resonant-state expansion including analytical mode normalization. For bound modes, our approach agrees with the previous theory, while it results in modified nonlinear pulse propagation for leaky modes. For instance, it predicts a more efficient generation of Stokes and anti-Stokes bands with an earlier onset than expected from the previous theory for bound modes. These effects have been demonstrated numerically for a gas-filled hollow-core annulus fiber that supports leaky modes, rendering conventional bound-mode theory inappropriate for such systems. Moreover, we find that leaky modes provide modulation instability, not only in the anomalous but also in the normal dispersion regime. The modulation instability can occur for all frequencies, which is a fundamental difference to bound modes.

Figure 1

(a) Cross section and schematics of an annulus fiber made from silica (${\mathrm{SiO}}_2$) and filled by xenon (Xe) in nitrogen (N) surrounding. The lower left inset shows energy-level diagram for four-wave mixing occurring due to the annihilation of two photons at frequencies ${\omega }_1$ and ${\omega }_2$ with creation of two new photons at frequencies ${\omega }_3$ and ${\omega }_4$. (b) Spatial distribution of the axial component of the real-valued Poynting vector of the fundamental leaky mode of the annulus waveguide geometry shown in (a). Outside the core region, a logarithmic scale is used for the distance to the fiber core. The white dashed line in (c) indicates the inner radius of the fiber. The refractive indices of silica and gases are taken from Refs. [10, 11]. The inner and outer radii of the annulus waveguide are $R_{\text{1}}=30\mu \mathrm{m}$ and $R_{\text{2}}=30.476\mu \mathrm{m}$, respectively (wavelength $\lambda =470$ nm).

Figure 2

Gain spectra of modulation instability at a pump wavelength of $\lambda =470$ nm for the annulus fiber with $\gamma =(8.47\times {10}^{-6}-4.58\times {10}^{-9}i){\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$, $\alpha =0.24{\mathrm{m}}^{-1}$, and $|{\overline{\beta }}^{\left(2\right)}|=6.84\times {10}^{-2}{\mathrm{ps}}^2{\mathrm{km}}^{-1}$. The red dashed line indicates the gain spectrum of modulation instability calculated in the anomalous dispersion regime with $\text{sgn}\left({\overline{\beta }}^{\left(2\right)}\right)=-1$, while the solid green line is obtained for $\text{sgn}\left({\overline{\beta }}^{\left(2\right)}\right)=1$, corresponding to the normal dispersion regime. Note that in both cases, there is a nonzero gain for all frequencies.

Figure 3

Spectral distributions of the various relevant parameters. (a) Real and (b) imaginary parts of the effective refractive index for the fundamental leaky mode of the annulus fiber shown in Fig. 1. Comparisons of results obtained by the resonant-state expansion (RSE) and the bound-mode theory (BMT) are given in panels (c) and (d), respectively, for the real and imaginary parts of the nonlinearity parameter of that mode. (e) Phase mismatch for degenerate four-wave mixing at a pump wavelength of $470\mathrm{nm}$ and initial powers of $3.1\mathrm{MW}$ (green solid line) and $3.2\mathrm{MW}$ (red dashed line), respectively. The central arrow indicates the position of pump (P) wave, while the side arrows indicate the estimated positions of the Stokes (S) and the anti-Stokes (AS) bands that occur at the wavelength with vanishing phase mismatch. (f) Second-order dispersion coefficient. All three waves are in the normal dispersion region. The gray vertical dashed lines in all panels indicate the spectral positions of the Stokes and the anti-Stokes waves.

Figure 4

Spatial-spectral evolution of the total power-spectral-density $S\left(\lambda \right)$ obtained by using our theory based on the resonant-state expansion and the bound-mode theory for an initial pump power of $3.1\mathrm{MW}$ [panels (a) and (b)] and $3.2\mathrm{MW}$ [panels (c) and (d)], respectively. Note that the total power spectral density $S\left(\lambda \right)$ is normalized to the initial power density. All other simulation parameters can be found in Ref. [55].

Figure 5

Normalized pump [panels (a) and (b)], Stokes [panels (c) and (d)], and anti-Stokes [panels (e) and (f)] power as a function of propagation distance. The results have been obtained by using our theory based on the resonant-state expansion (red solid line) and the bound-mode theory (blue dashed line) for initial pump powers of 3.1 MW (left column) and 3.2 MW (right column).