Efficient broadband frequency conversion via shortcut to adiabaticity

TL;DR

The paper addresses robust, broadband frequency conversion in nonlinear crystals by applying shortcut-to-adiabaticity (STA) techniques. It compares counter-diabatic driving and Lewis-Riesenfeld invariant–based inverse engineering for sum-frequency generation (SFG) in an aperiodically poled crystal and demonstrates that STA can drastically reduce crystal length while improving robustness to wavelength and temperature fluctuations. CD driving enables near-adiabatic performance in short devices but requires complex poling and large extra coupling, while LR invariant with LZ optimization offers a practical compromise with a constant coupling, achieving robust conversion in moderate lengths. The work provides design principles for STA-based nonlinear frequency conversion and highlights implications for fabrication and applications in broadband optical mixing.

Abstract

The method of adiabatic frequency conversion, in analogy with the two level atomic system, has been put forward recently and verified experimentally to achieve robust frequency mixing processes such as sum and difference frequency generation. Here we present a comparative study of efficient frequency mixing using various techniques of shortcuts to adiabaticity (STA) such as counter-diabatic driving and invariant-based inverse engineering. We show that, it is possible to perform sum frequency generation by properly designing the poling structure of a periodically poled crystal and the coupling between the input lights and the crystal. The required crystal length for frequency conversion is significantly decreases beyond the adiabatic limit. Our approach significantly improves the robustness of the process against the variation in temperature as well as the signal frequency. By introducing a single parameter control technique with constant coupling and combining with the inverse engineering, perturbation theory and optimal control, we show that the phase mismatch can be further optimized with respect to the fluctuations of input wavelength and crystal temperature that results into a novel experimentally realizable mixing scheme.

Figures (6)

• Figure 1: Schematic of controlled aperiodic structure of a poled crystal with designed continuous variation of phase mismatch $\Delta K(z)$ along the direction of propagation for realizing shortcuts to adiabatic sum frequency conversion, where $\omega_1$, $\omega_2$ are the signal and pump frequencies respectively whereas $\omega_1 + \omega_2 = \omega_3$ represents the idler frequency.
• Figure 2: Conversion of modes along the direction of propagation of the crystal, where (a) incomplete conversion with adiabatic condition is violated with $I_P = 60~ \text{MW}/\text{cm}^2$ and (b) conversion is complete as the adiabatic condition is satisfied with $I_P = 360 ~ \text{MW}/\text{cm}^2$, (c) and (d) respective Bloch vector trajectories.
• Figure 3: (a) Profile of required $\Delta K_{eff}$ for the application of CD driving for different crystal lengths, i.e., $L = 20~\text{mm}$ (dotted red), $L = 2 ~\text{mm}$ (dashed blue), $L = 0.2~\text{mm}$ (dot-dashed magenta), and adiabatic one $L = 200 ~\text{mm}$ (solid black). (b) Nature of additional coupling required for complete mode transfer in different lengths. (c) Conversion of modes along the direction of propagation of the crystal using the CD driving for crystal length $2 mm$ with $I_P = 60~ \text{MW}/\text{cm}^2$, and (d) respective Bloch vector trajectory.
• Figure 4: (a) $z$ dependence of $\zeta$ (solid red) and $\beta$ (dashed blue), obtained from Eqs. (\ref{['gam']}) and (\ref{['beta']}), with the boundary conditions as satisfied by the eigenstates of LR invariant i.e., $\zeta(0) = 0$ and $\zeta(L) = \pi$. (b) The coupling constant $Q_0$ (solid red) and optimal $\Delta K_{opt}$ (dashed blue) designed from Eq. (\ref{['optDel']}) by using the LZ optimization of the LR invariant engineering. (c) The corresponding conversion of modes along the direction of propagation of the crystal and (d) respective Bloch vector trajectory. Parameters: $c_1=-1.47$ for crystal length $2~\text{mm}$ with $I_P = 360~\text{MW}/\text{cm}^2$.
• Figure 5: Conversion efficiency of modes with respect to the variation of signal wavelength for different pump intensities with $I_P = 10~\text{MW}/\text{cm}^2$ (blue dashed), $I_P = 60~\text{MW}/\text{cm}^2$ (solid red) and $I_P = 360~\text{MW}/\text{cm}^2$ (black dashed-dotted ) in (a), (b) and (c); and for different crystal length with pump intensity $I_P = 360~\text{MW}/\text{cm}^2$, $L=2~\text{mm}$ (blue dashed), $L = 10 ~\text{mm}$ (solid red), $L = 20~\text{mm}$ (black dashed-dotted) in (d), (e) and (f) respectively. Here for comparison, (a,d) present adiabatic SFG, (b,e) presents CD driving, and (c,f) presents the optimal SFG designed by inverse engineering.