On-demand generation of propagation-invariant photons with orbital angular momentum

We study the generation of propagation invariant photons with orbital angular momentum by spontaneous parametric down conversion (SPDC) using a Bessel-Gauss pump beam. The angular and conditional angular spectra are calculated for an uniaxial crystal optimized for type I SPDC with standard Gaussian pump beams. It is shown that, as the mean value of the magnitude of the transverse wave vector of the pump beam increases, the emission cone is deformed into two non-coaxial cones that touch each other along a line determined by the orientation of the optical axis of the nonlinear crystal. At this location, the conditional spectrum becomes maximal for a pair of photons, one of which is best described by a Gaussian-like photon with a very small transverse wave vector, and the other a Bessel-Gauss photon with a distribution of transverse wave vectors similar in amplitude to that of the incident pump beam. A detailed analysis is then performed of the angular momentum content of SPDC photons by the evaluation of the corresponding transition amplitudes. As a result, we obtain conditions for the generation of heralded single photons which are approximately propagation invariant and have orbital angular momentum. A discussion is given about the difficulties in the interpretation of the results in terms of conservation of optical orbital angular momentum along the vector normal to the crystal surface. The angular spectra and the conditional angular spectra are successfully compared with available experimental data recently reported in the literature.

Figure 1

Angular spectrum (AS) for a pump beam with a waist $W=0.0007\mu$m${}^{-1}$ and different values of transverse wave number: (A) ${\kappa }_{\perp }^p=0.05\mu$m${}^{-1}$; (B) ${\kappa }_{\perp }^p=0.09\mu$m${}^{-1}$; (C) ${\kappa }_{\perp }^p=0.15\mu$m${}^{-1}$. (a) and (b) consider a 1 mm long BBO crystal while in (c) and (d) the crystal length is increased to 2 mm. In (a) and (c), the AS is calculated numerically through Eq. (9), while in (b) and (d) the AS is calculated from the analytic expression Eq. (23). The optical axis of the crystal is located in the $Y$-$Z$ plane $\mathbf{a}\sim (0,0.49,0.87)$. For normal incidence the resulting walk-off angle is ${\rho }_{wo}\sim -\beta {\mathrm{a}}_y\sim 0.068\mathrm{rad}$ while the emission cones have a aperture angles around ${\theta }_A\sim r_{\mathrm{AS}}/k_z^s\sim 0.034\mathrm{rad}$, $r_{\mathrm{AS}}=\sim 0.49\mu$m${}^{-1}$. The estimated transverse radii of the major emission cones are $R_{+}^A\sim 0.51\mu {\mathrm{m}}^{-1}$, $R_{+}^B\sim 0.53\mu {\mathrm{m}}^{-1}$, and $R_{+}^C\sim 0.56\mu {\mathrm{m}}^{-1}$, while for the minor emission cones are $R_{-}^A\sim 0.42\mu$m${}^{-1}$,$R_{-}^B\sim 0.36\mu$m${}^{-1}$, and $R_{-}^C\sim 0.27\mu$m${}^{-1}$; their centers are located at $A_{\pm }^A\sim \pm 0.02\mu$m${}^{-1}$, $A_{\pm }^B\sim \pm 0.04\mu$m${}^{-1}$, and $A_{\pm }^C\sim \pm 0.07\mu$m${}^{-1}$.

Figure 2

Maximal conditional angular spectrum for a pump Bessel Gauss beam with waist $W=0.0007\mu$m${}^{-1}$ in (a) and (b) and $W=0.007\mu$m${}^{-1}$ in (c).The transverse wave numbers are: (A) ${\kappa }_{\perp }^p=0.05\mu$m${}^{-1}$, (B) ${\kappa }_{\perp }^p=0.09\mu$m${}^{-1}$, and (C) ${\kappa }_{\perp }^p=0.15\mu$m${}^{-1}$. The crystal length is 1 mm in (a) and (c) and 2 mm in (b). The optical axis of the crystal is located in the $Y$-$Z$ plane $\mathbf{a}\sim (0,0.49,0.87).$

Figure 3

Schematic picture of the SPDC process involving propagation invariant pump, signal, and idler photons with orbital angular momentum. The emission cone is expected to have a radii given by $r_{\mathrm{AS}}={(n_o\omega /c)}^2(1-n_{\mathrm{eff}}/n_o)/2$. The main propagation axis of the signal and idler photons will be approximately located on that cone.

Figure 4

Modulus of the transition element $F$, Eq. (45), as a function of the orbital angular momentum of the signal ${\ell }^s$ and idler ${\ell }^i$ photon. They involve a pump Bessel-Gauss photon with transverse wave number ${\kappa }_{\perp }^p=0.01\mu$m${}^{-1}$, a signal Bessel photon with ${\kappa }_{\perp }^s=0.0001\mu$m${}^{-1}$, and an idler photon with ${\kappa }_{\perp }^i=0.01\mu$m${}^{-1}$. The idler and signal photons are emitted with their main propagation axis with orientation angles $\tilde{\theta }={\theta }_A$, ${\tilde{\phi }}_s=-\pi /2$, and ${\tilde{\phi }}_i=\pi /2$ for the first row, while ${\tilde{\phi }}_s=0$ and ${\tilde{\phi }}_i=\pi$ in the second row. The pump angular quantum number is ${\ell }^p=0$ in (a) and (d), ${\ell }^p=1$ in (b) and (e), and ${\ell }^p=2$ in (c) and (f). The BBO crystal length is 1 mm, its optical axis is located in the $Y$-$Z$ plane, and the width of the transversal wave number for the Bessel-Gauss photons is $W=0.0005\mu$m${}^{-1}$. The optical axis of the crystal is located in the $Y$-$Z$ plane $\mathbf{a}\sim (0,0.49,0.87).$

Figure 5

Modulus of the transition element $F$, Eq. (45), as a function of the orbital angular momentum ${\ell }^s$ of the signal and ${\ell }^i$ of the idler photon. They involve a pump Bessel-Gauss photon with transverse wave number ${\kappa }_{\perp }^p=0.05\mu$m${}^{-1}$, a signal Bessel photon with ${\kappa }_{\perp }^s=0.001\mu$m${}^{-1}$, and an idler photon with ${\kappa }_{\perp }^i=0.05\mu$m${}^{-1}$. The idler and signal photons are emitted with their main propagation axis with orientation angles $\tilde{\theta }={\theta }_A$, ${\tilde{\phi }}_s=-\pi /2$, and ${\tilde{\phi }}_i=\pi /2$ in the first row, while ${\tilde{\phi }}_s=0$ and ${\tilde{\phi }}_i=\pi$ in the second row, The pump angular quantum number is ${\ell }^p=0$ in (a) and (d), ${\ell }^p=1$ in (b) and (e) and ${\ell }^p=2$ in (c) and (f). The BBO crystal length is 1 mm, its optical axis is located in the $Y$-$Z$ plane and the width of the transversal wave number for the Bessel-Gauss photons is $W=0.0005\mu$m${}^{-1}$. The optical axis of the crystal is located in the $Y$-$Z$ plane $\mathbf{a}\sim (0,0.49,0.87)$.

Figure 6

Marginal distribution of orbital angular momentum of the signal ${\ell }^s$ and idler ${\ell }^i$ photon. They involve a pump Bessel-Gauss profile with transverse wave number ${\kappa }_{\perp }^p=0.01\mu$m${}^{-1}$, a signal Bessel photon with ${\kappa }_{\perp }^s=0.0001\mu$m${}^{-1}$, and an idler photon with ${\kappa }_{\perp }^i=0.01\mu$m${}^{-1}$. The idler and signal photons are emitted with their main propagation axis with orientation angles $\tilde{\theta }={\theta }_{ec}$, ${\tilde{\phi }}_s=-\pi /2$, and ${\tilde{\phi }}_i=\pi /2$. The pump angular quantum number is ${\ell }^p=0$ in (a) and (d), ${\ell }^p=1$ in (b) and (e), and ${\ell }^p=2$ in (c) and (f). The BBO crystal length is 1 mm, its optical axis is located in the $Y$-$Z$ plane and the width of the transversal wave number for the Bessel-Gauss photons is $W=0.0005\mu$m${}^{-1}$. The optical axis of the crystal is located in the $Y$-$Z$ plane $\mathbf{a}\sim (0,0.49,0.87).$