Third-order spontaneous parametric down-conversion in thin optical fibers as a photon-triplet source

We study the third-order spontaneous parametric down-conversion (TOSPDC) process, as a means to generate entangled photon triplets. Specifically, we consider thin optical fibers as the nonlinear medium to be used as the basis for TOSPDC in configurations where phase matching is attained through the use of more than one fiber transverse modes. Our analysis in this paper, which follows from our earlier paper [Opt. Lett. 36, 190–192 (2011)], aims to supply experimentalists with the details required in order to design a TOSPDC photon-triplet source. Specifically, our analysis focuses on the photon triplet state, on the rate of emission, and on the TOSPDC phase-matching characteristics for the cases of frequency-degenerate and frequency nondegenerate TOSPDC.

Figure 1

(a) Frequency-degenerate phase matching for TOSPDC at 1.350 $\mu$m (blue line), 1.596 $\mu$m (black line), and 1.800 $\mu$m (red line). (b) Degenerate TOSPDC wavelength and nonlinear coefficient $\gamma$ vs. phase-matching radius.

Figure 2

Phase-matched emission frequencies plotted as a function of the pump frequency, for different fiber radii, and assuming the following idler wavelengths, kept constant for each of the four panels: (a) ${\lambda }_i=0.6$ $\mu$m, (b) ${\lambda }_i=0.8$ $\mu$m, (c) ${\lambda }_i=1.2$ $\mu$m, and (d) ${\lambda }_i=1.6$ $\mu$m.

Figure 3

Nonlinear coefficient $\gamma$ as a function of ${\omega }_p$ and $\Delta$ for $r=0.395$ $\mu$m and ${\omega }_i=2\pi c/1.596$ $\mu \text{m}$. The black-solid line represents frequency combinations leading to perfect phase matching.

Figure 4

(a) Phase-matched emission frequencies as a function of the core radius, for a fixed pump wavelength (${\lambda }_p=0.532$ $\mu \text{m}$). (b) Phase-matched emission frequencies as a function of the pump frequency, for a fixed fiber radius ($r=0.395$ $\mu$m).

Figure 5

Representation of the JSI for our frequency-degenerate TOSPDC source design, plotted as a function of the frequency variables ${\nu }_A$ and ${\nu }_B$ for the following fixed values of ${\nu }_{+}$: ${\nu }_{+}=15\times {10}^9$rad/s (a), ${\nu }_{+}=0$ (b), and ${\nu }_{+}=-15\times {10}^9$rad/s (c). (d) JSI plotted as a function of ${\nu }_{+}$ for ${\nu }_A={\nu }_B=0$.

Figure 6

Plotted as function of the three emitted frequencies $\{{\omega }_r,{\omega }_s,{\omega }_i\}$: (a) Phase-matching function $|\phi ({\omega }_r,{\omega }_s,{\omega }_i){|}^2$, (b) pump spectral amplitude $|\alpha ({\omega }_r+{\omega }_s+{\omega }_i){|}^2$, and (c) JSI. (d) Two-photon spectrum $I_2({\omega }_r,{\omega }_s)$. (e) Single-photon spectrum $I_1\left({\omega }_r\right)$. Panels (a)–(c) are similar to a figure from Ref. [19].

Figure 7

Frequency-non-degenerate TOSPDC photon-triplet state. (a)–(c) Two-photon spectra obtained by integrating the JSI over each of the three emission frequencies in turn. The light shaded bands indicate spectral filtering used. (d)–(f) Single photon spectra for each of the three emission modes, including the effect of spectral filtering.

Figure 8

Emitted flux as a function of the pump bandwidth for the following cases: (a) Frequency-degenerate TOSPDC source, evaluated from Eq. (19) (blue dots); (b) Frequency-non-degenerate TOSPDC source, evaluated from Eq. (19) (red dots); and frequency-non-degenerate TOSPDC from the closed analytic expression, i.e., Eq. (28) (black solid line). Values obtained for the monochromatic pump limit, through Eq. (22), are indicated by green dots.

Figure 9

Emitted flux as a function of the fiber length for the following cases: (a) frequency-degenerate TOSPDC source, evaluated from Eq. (19) (blue dots); (b) frequency-non-degenerate TOSPDC source, evaluated from Eq. (19) (red dots), and frequency-non-degenerate TOSPDC from closed analytic expression, i.e., Eq. (28) (black solid line).

Figure 10

Emitted flux as a function of the pump power for (a) the frequency-degenerate TOSPDC source, evaluated from Eq. (19) (blue dots) and (b) the frequency-non-degenerate TOSPDC source, evaluated from Eq. (19) (red dots), and from closed analytic expression, i.e., Eq. (28) (solid black line).