Emission of orbital-angular-momentum-entangled photon pairs in a nonlinear ring fiber utilizing spontaneous parametric down-conversion

We suggest the generation of photon pairs in a thermally induced nonlinear periodically poled silica fiber by spontaneous parametric down-conversion. Photons are generated directly in eigenstates of optical angular momentum. Photons in a pair can be entangled in these states as well as in frequencies. We identify suitable spatial and polarization modes giving an efficient nonlinear interaction. By changing the pump field properties both narrow- and broadband down-converted fields can be obtained.

Figure 1

(a) Sketch of a ring fiber with two small poling holes and (b) radial profiles of indices of refraction $n$ at the pump $({\lambda }_p^0=0.775$ $\mu \text{m})$ and signal $({\lambda }_s^0=1.55$ $\mu \text{m})$ wavelengths.

Figure 2

(a) Effective refractive index $n_{p,\mathrm{eff}}$ of the pump field in dependence on azimuthal number $n_p$ for ${\lambda }_p^0=0.775\mu \text{m}$. In (b), detail of the graph around $n_p=0$ is shown.

Figure 3

(a) Effective refractive index $n_{s,\mathrm{eff}}$ of the signal field in dependence on azimuthal number $n_s$ for ${\lambda }_s^0=1.55\mu \text{m}$. In (b), detail of the graph around $n_s=0$ is shown.

Figure 4

Amplitude and phase of components ${\mathbf{e}}_x(x,y)$ and ${\mathbf{e}}_z(x,y)$ of electric-field amplitudes for modes ${\mathrm{TE}}_{01}$ (a), ${\mathrm{TM}}_{01}$ (b), ${\mathrm{HE}}_{11,R}$ (c), and ${\mathrm{HE}}_{21,R}$ (d) for the signal field at ${\lambda }_s^0=1.55\mu \text{m}$; $x=rcos\left(\theta \right)$ and $x=rsin\left(\theta \right)$. The Cartesian $x$ and $y$ axes' labels are in $\mu \text{m}$ and the components are normalized according to $\int dxdy|{\mathbf{e}}_{x,z}{(x,y)|}^2=1$.

Figure 5

Probabilities $p_l$ of measuring an OAM eigenmode $t_l$ for the $x$ and $z$ components of electric-field amplitude ${\mathbf{e}}_{\eta }(r,\theta ,\omega )$ for modes ${\mathrm{TE}}_{01}$ (a), ${\mathrm{TM}}_{01}$ (b), ${\mathrm{HE}}_{11,R}$ (c), and ${\mathrm{HE}}_{21,R}$ (d) for the signal field at ${\lambda }_s^0=1.55\mu \text{m}$.

Figure 6

Phase mismatch $\Delta \beta$ for nonlinear processes $({\mathrm{HE}}_{21}^p,{\mathrm{HE}}_{21}^s,{\mathrm{HE}}_{11}^i)$, $({\mathrm{HE}}_{21}^p,{\mathrm{TE}}_{01}^s,{\mathrm{HE}}_{11}^i)$, and $({\mathrm{HE}}_{21}^p,{\mathrm{TM}}_{01}^s,{\mathrm{HE}}_{11}^i)$. The gray horizontal pattern describes spatial spectrum ${\tilde{\chi }}^{\left(2\right)}\left(\beta \right)$ of a rectangular QPM grating with $\Lambda =42.9\mu \text{m}$ and $L=10$ cm.

Figure 7

Absolute value $|{\mathbf{e}}_x|$ of the $x$ component of electric-field amplitude depending on radius $r$ for pump mode ${\mathrm{HE}}_{21}^p$, signal mode ${\mathrm{HE}}_{21}^s$, and idler mode ${\mathrm{HE}}_{11}^i$. Normalization is such that $\int dxdy|{\mathbf{e}}_x{(x,y)|}^2=1$.

Figure 8

Spectral photon-number density $N_s$ created by modes ${\mathrm{HE}}_{21,R},{\mathrm{HE}}_{11,R},{\mathrm{HE}}_{11,L},{\mathrm{TE}}_{01}$, and ${\mathrm{TM}}_{01}$ in dependence on wavelength ${\lambda }_s$; $N_s\left({\omega }_s\right)={\sum }_{{\eta }_p,{\eta }_s,{\eta }_i}{N}_{s,{\eta }_s,{\eta }_i}^{{\eta }_p}\left({\omega }_s\right)$.

Figure 9

Phase mismatch $\Delta \beta$ for nonlinear processes pumped by mode ${\mathrm{HE}}_{11,R}^p$ with signal and idler fields in modes ${\mathrm{HE}}_{21},{\mathrm{TE}}_{01}$, and ${\mathrm{TM}}_{01}$ in dependence on wavelength of signal photon ${\lambda }_s$. The gray horizontal pattern describes spatial spectra ${\tilde{\chi }}^{\left(2\right)}\left(\Delta \beta \right)$ of rectangular nonlinear modulation with $\Lambda =41.06\mu \text{m}$ (upper pattern) and $\Lambda =42.28\mu \text{m}$ (lower pattern); $L=10$ cm.

Figure 10

Spectral photon-number density $N_s$ originating in nonlinear process $({\mathrm{HE}}_{11,R}^p,{\mathrm{TE}}_{01}^s,{\mathrm{TE}}_{01}^i)$; $\Lambda =42.28\mu \text{m}$ and $L=10$ cm in dependence on wavelength ${\lambda }_s$.

Figure 11

Probability density $p_{t,i},p_{t,i}=\mathcal{C}{\left|\tilde{\Phi }(0,t_i)\right|}^2$, as a function of idler-photon detection time $t_i$ for a signal photon detected at time $t_s=0$ s for processes $({\mathrm{HE}}_{21,R}^p,{\mathrm{HE}}_{21,R}^s,{\mathrm{HE}}_{11,R}^i)$ $(\Lambda =42.9\mu \text{m}$) and $({\mathrm{HE}}_{11,R}^p,{\mathrm{TE}}_{01}^s,{\mathrm{TE}}_{01}^i)$ $(\Lambda =42.28\mu \text{m}$); $L=10$ cm. Constant $\mathcal{C}$ is defined such that ${\int }_{-\infty }^{\infty }d{t}_i{p}_{t,i}\left(t_i\right)=1$.

Figure 12

Spectral photon-number density $N_s$ arising from nonlinear processes $({\mathrm{HE}}_{11,R}^p,{\mathrm{HE}}_{21,R}^s,{\mathrm{HE}}_{21,L}^i)$ and $({\mathrm{HE}}_{11,R}^p,{\mathrm{HE}}_{21,L}^s,{\mathrm{HE}}_{21,R}^i)$ in dependence on wavelength ${\lambda }_s$. The curves nearly coincide; $\Lambda =41.06\mu \text{m}$ and $L=10$ cm.

Figure 13

Cut of absolute value $|{\Phi }_{1_s,-1_i}({\omega }_s,{\omega }_i)|$ of two-photon spectral amplitude along the line (a) ${\omega }_s={\omega }_i-{\omega }_i^0+{\omega }_s^0$ and (b) ${\omega }_s={\omega }_p^0-{\omega }_i$ appropriate for the process $({\mathrm{HE}}_{11,R}^p,{\mathrm{HE}}_{21,R}^s,{\mathrm{HE}}_{21,L}^i)$ pumped by a pulsed field; ${\sigma }_p=0.85$ nm, $\Lambda =41.06\mu \text{m}$, and $L=10$ cm. It holds that $\int d{\omega }_{s}d{\omega }_i{\left|\Phi ({\omega }_s,{\omega }_i)\right|}^2=1$.

Figure 14

Number $K_{\omega }$ of independent spectral modes as it depends on pump-field spectral width ${\sigma }_p$; $\Lambda =41.06\mu \text{m}$ and $L=10$ cm.