Adiabatic geometric phase in fully nonlinear three-wave mixing

In a nonlinear three-wave mixing process, the interacting waves can accumulate an adiabatic geometric phase (AGP) if the nonlinear coefficient of the crystal is modulated in a proper manner along the nonlinear crystal. This concept was studied so far only for the case in which the pump wave is much stronger than the two other waves, hence can be assumed to be constant. Here we extend this analysis for the fully nonlinear process, in which all three waves can be depleted and we show that the sign and magnitude of the AGP can be controlled by the period, phase, and duty cycle of the nonlinear modulation pattern. In this fully nonlinear interaction, all the states of the system can be mapped onto a closed surface. Specifically, we study a process in which the eigenstate of the system follows a circular rotation on the surface. Our analysis reveals that the AGP equals to the difference between the total phase accumulated along the circular trajectory and that along its vertical projection, which is universal for the undepleted (linear) and depleted (nonlinear) cases. Moreover, the analysis reveals that the AGPs in the processes of sum-frequency generation and difference-frequency generation have opposite chirality. Finally, we utilize the AGP in the fully nonlinear case for splitting the beam into different diffraction orders in the far field.

Figure 1

(a) A typical example of a clockwise circular rotation of the QPM vector $\mathbf{B}\left(\tau \right)$ on its parameter space sphere with $\mathrm{\Theta }=\pi /4$. (b) The variations of the poling period $\mathrm{\Delta }\mathrm{\Gamma }$, the duty cycle $\mathrm{\Xi }$, and phase ${\varphi }_{\mathrm{d}}$, of the nonlinear coefficient along the nonlinear crystal in order to achieve $\mathrm{\Theta }=\pi /4$. (c) A corresponding rotation of the state vector $\mathbf{W}\left(\tau \right)$, which follows the clockwise rotation of the QPM vector in (a), draws a trajectory onto the state surface (7) with initial conditions $(I_1,I_2,I_3)=(0.4,0.6,0)$ (where $I_j={\left|q_j^0\right|}^2$). (d) AGP accumulations for $q_{1,2}$ in (c) along the scaled propagation coordinate.

Figure 2

(a) Interferometric setup for measuring the phase difference between clockwise and counterclockwise circular rotations. (b) Circular rotation for SFG. The blue and the red trajectories show the counterclockwise (c.c.) and clockwise (c.) rotations, respectively. The black trajectory is the round-trip (r.t.) motion. Here, we select $\mathrm{\Theta }=0.3\pi$ and $(I_1,I_2,I_3)=(0.4,0.6,0)$. (c) Accumulation of ${\beta }_1$ for the clockwise (red solid curve) and counterclockwise (blue dashed curve) in (b).

Figure 3

Circular rotation for SFG. (a) ${\beta }_2/{\beta }_1$ versus $I_1/I_2$ with different values of $\mathrm{\Theta }$. (b)${\beta }_{1,2}$ versus $I_1-I_2$ for $\mathrm{\Theta }=\pi /3$. (c) The sum of AGP for $q_{1,2}$ versus $\mathrm{\Theta }$ with a different depletion level. Here, the symbols (circle, square, and rhombus) represent the numerical results from the interferometric scheme in Eq. (14), while the dashed curve is the prediction from Eqs. (19) and (21). The black curve is the theoretical result from Eq. (15) for the undepleted case.

Figure 4

Circular rotation in the case of DFG. (a) The blue and red trajectories are the counterclockwise (c.c.) and clockwise (c.) rotations, respectively, with $\mathrm{\Theta }=\pi /3$ for initial conditions $(I_1,I_2,I_3)=(0,0.5,0.5)$. The black curve represents the round-trip case. (b) The AGP for $q_3$ (i.e., ${\beta }_3$) as a function of $\mathrm{\Theta }$ for different ratios $I_2/I_3$. The symbols were calculated with Eq. (14). The dashed lines were calculated with Eq. (21). The black solid curve represents the undepleted case.

Figure 5

Schematic drawing of the grating simulation. Each segment of poling in the $x$ direction corresponds to clockwise rotation and round-trip motion in parameter space. The period of each segment is set to be $\mathrm{\Lambda }=50\mu \mathrm{m}$.

Figure 6

(a) Phase profile of the beam at the end of the geometric phase grating. (b) Far-field intensity after the crystal end.

Figure 7

(a) Trajectory of the $\mathbf{W}\left(\tau \right)$ of clockwise rotation with $\mathrm{\Theta }=0.4\pi$ in the full depleted case (i.e., $I_1=I_2$). (b)The evolutions of the phase of ${\beta }_1$ for the clockwise and counterclockwise rotation, which are calculated via Eq. (21).