Four-photon scattering in birefringent fibers

Four-photon scattering in nonlinear waveguides is an important physical process that allows photon-pair generation in well-defined guided modes, with high rate and reasonably low noise. Most of the experiments to date used the scalar four-photon scattering process in which the pump photons and the scattered photons have the same polarization. In birefringent waveguides, vectorial four-photon scatterings are also allowed: these vectorial scattering processes involve photons with different polarizations. In this paper, the theory of four-photon scattering in nonlinear, birefringent, and dispersive fibers is developed in the framework of the quantum theory of light. The work focuses on the spectral properties and quantum correlations (including entanglement) of photon pairs generated in high-birefringence and low-birefringence fibers.

Figure 1

Photon-flux spectral densities $f_x\left(L,\Omega \right)$ for the (a) anomalous- and (b) normal-dispersion regimes. The first-order approximation $f_x\left(L,\Omega \right)={\left|{\xi }_{xx}\left(L,\Omega \right)\right|}^2/\left(2\pi \right)$ (dashed black lines) is compared with the exact value (solid gray lines) obtained by solving Eqs. (27) in the Heisenberg picture. The parameters used in this figure are $\gamma =3/\left(\text{W}\text{km}\right)$, $P_{0x}=300\text{mW}$, and ${\beta }_2=\pm 20{\text{ps}}^2/\text{km}$ [$-$ in panel (a) and $+$ in panel (b)], and $L$ takes successively the values 100, 200, and 300 m.

Figure 2

Photon-flux spectral densities (a) $f_x\left(L,\Omega \right)$ and (b) $f_y\left(L,\Omega \right)$ as functions of $\Omega$ for propagation distances $L=100\text{m}$, $L=200\text{m}$, and $L=300\text{m}$. The other parameters used in this figure are $\gamma =3/\left(\text{W}\text{km}\right)$, $P_{0x}=P_{0y}=150\text{mW}$, $\Delta {\beta }_1=200\text{ps}/\text{km}$, and ${\beta }_2=\pm 15{\text{ps}}^2/\text{km}$. Light gray and dark gray curves correspond to the normal $\left({\beta }_2=+15{\text{ps}}^2/\text{km}\right)$ and anomalous dispersions $\left({\beta }_2=-15{\text{ps}}^2/\text{km}\right)$, respectively.

Figure 3

Photon-flux spectral densities $f_x\left(L,\Omega \right)$ (light gray) and $f_y\left(L,\Omega \right)$ (dark gray) as functions of $\Omega$ for propagation distances $L=15\text{cm}$, $L=30\text{cm}$, and $L=45\text{cm}$. The other parameters used in this figure are $\gamma =36/\left(\text{W}\text{km}\right)$, $P_{0x}=P_{0y}=20\text{W}$, $\Delta {\beta }_1=400\text{ps}/\text{km}$, and ${\beta }_2=-139{\text{ps}}^2/\text{km}$ $\left(\alpha =-1.25\right)$.

Figure 4

Photon-flux spectral densities on the slow (gray lines) and fast (black lines) axes of the fiber as a function of $\Omega$. In panel (a), the dispersion is normal and the pump is polarized along the slow axis of the fiber. In panel (b), the situation is opposite: the dispersion is anomalous and the polarization of the pump wave is along the fast axis. The parameters used in this figure are $\gamma =3/\left(\text{W}\text{km}\right)$, $P_0=1\text{W}$, $\left|\Delta {\beta }_0\right|=2{\text{m}}^{-1}$, and $\left|{\beta }_2\right|=5{\text{ps}}^2/\text{km}$. Three propagation distances ($L=50\text{m}$, $L=100\text{m}$, and $L=150\text{m}$) are considered.

Figure 5

Graphical representation of the mechanism leading to the growth of the number of photon pairs in the field in the case of a scalar scattering. Each arrow represents a scattering event, i.e., the creation (right-pointing arrow) or the annihilation (left-pointing arrow) of a signal-idler pair. Column labels represent the number of photon pairs in the field. Row labels indicate the order in Dyson’s perturbation series accounting for the scattering events represented by arrows in that row.