Gaussian-beam-propagation theory for nonlinear optics involving an analytical treatment of orbital-angular-momentum transfer

We present a general, Gaussian spatial-mode propagation formalism for describing the generation of higher-order multi-spatial-mode beams generated during nonlinear interactions. Furthermore, to implement the theory, we simulate optical angular momentum transfer interactions and show how one can optimize the interaction to reduce the undesired modes. Past theoretical treatments of this problem have often been phenomenological, at best. Here we present an exact solution for the single-pass no-cavity regime, in which the nonlinear interaction is not overly strong. We apply our theory to two experiments, with very good agreement, and give examples of several more configurations, easily tested in the laboratory.

Figure 1

LG mode intensity profiles for (a) $\ell =0$ and increasing $p$, (b) $p=0$ and increasing $\ell$, and (c) several superpositions of $\ell$ with $p=0,1,$ and 2. Due to interference, superpositions with $\ell$ of opposite sign will have a symmetric petallike structure with $|{\ell }_1|+|{\ell }_2|$ petals. See Ref. [2] for an excellent description of optical OAM origins and behavior.

Figure 2

In (a) we depict the simple stimulated-down-conversion scheme of our simulation. In (b) through (d) we show the input beam profiles of the pump and signal beams, a histogram giving the mode structure of the output (idler) beam, and the spatial profile of the output (idler) beam from left to right. In (b) we use a $u_{0,0}$ signal as a baseline, and then in (c) and (d), we show how the signal beam can be chosen to tailor a clean higher OAM superposition in the idler mode.

Figure 3

In (a) we depict a nondegenerate four-wave mixing scheme, which occurs in ${}^{87}\mathrm{Rb}$. In (b) through (f), we study the result of this interaction for different pump profiles; from left to right the columns correspond to the profile of the pump beams, a histogram giving the mode structure of the conjugate beam, the spatial profile of the conjugate beam, and an experimental realization from Ref. [13]. Better qualitative agreement is observed when ${\omega }_s$ is allowed to take on any mode conserving OAM. This result suggests that, contrary to previous assumptions, the long-wavelength signal beam can, in fact, carry OAM.

Figure 4

In (a) and (c) we depict the two simple second-harmonic-generation schemes of our simulation. In (b) and (d) we show the input-beam profile, which is supplying the two pump photons; a histogram giving the mode structure of the output beam; and the spatial profile of the output beam from left to right. In (b) both pump photons come from a $u_{1,0}+u_{-1,0}$ superposition. In (c) and (d), a phase shift and a new geometry are chosen such that destructive interference cancels the response at $\ell =0$ [seen in (b)]. The pump photons are in the far infrared at 1140 nm, and thus, the second-harmonic response is at 570 nm.

Figure 5

In (a) and (c) we depict the two simple third-harmonic-generation schemes of our simulation. In (b) and (d) we show the input-beam profiles, which are supplying the three pump photons; a histogram giving the mode structure of the output beam; and the spatial profile of the output beam from left to right. In (b) all pump photons come from a $u_{1,0}+u_{-1,0}$ superposition. In (c) and (d), phase shifts and a new geometry are chosen such that destructive interference cancels the response at $\ell =\pm 1$ as seen in (b). The pump photons are in the far infrared at 1140 nm, and thus, the third-harmonic response is at 380 nm.

Figure 6

In (a) we depict the simple degenerate four-wave mixing scheme of our simulation. In (b) through (d) we show the pump-beam profiles, the signal-beam profile, a histogram giving the mode structure of the output (conjugate) beam, and the spatial profile of the output (conjugate) beam from left to right. In (b) the pump beams are of the form $u_{1,0}+u_{-2,0}$ and $u_{1,0}+e^{i\pi }{u}_{-2,0}$, and the signal beam is a $u_{0,0}$ mode. In (c) the pump beams remain the same, and the signal beam is a $u_{-1,0}$ mode. In (d) the pumps stay the same, but the signal beam is changed to $u_{2,0}$. This is a degenerate scheme, and all beams are 650 nm.

Figure 7

Geometry of the two-beam coordinate systems, which are separated in the $x\text{−}z$ plane by the angle $\theta$. Any point $P$ can be described by the cylindrical coordinates ($r,\varphi ,z$) or ($r^{\prime },{\varphi }^{\prime },z^{\prime }$). We associate the generated beam with the unprimed coordinate system and the input beam with the primed coordinate system; then using geometry, we deduce the form of $r^{\prime },{\varphi }^{\prime },z^{\prime }$ in terms of $r,\varphi ,z,$ and $\theta$.