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Journal of the Optical Society of America B

Optimal sum-frequency generation processes

Haifei Zhu, Yonggang Xu, Zhanrong Zhou, Yongfang Li

Published Feb 4, 2026• 0 citations
Publisher

Abstract

Sum-frequency generation in nonlinear optics is a key technology enabling optical frequency conversion. However, how to most effectively enhance the frequency conversion efficiency in nonlinear crystals remains a fundamental problem; shorter crystal length and lower pump intensity while maintaining high-efficiency frequency conversion are critical for enhancing device integration potential and improving device performance in practical sum-frequency generation (SFG) device design. This study employed the Pontryagin maximum principle to investigate the SFG process. A cost functional was formulated that complies with the minimum constraint during SFG, enabling the derivation of optimal control parameters. A central finding revealed an inverse proportionality between the coupling coefficient and the crystal length: for a given crystal length, the pump intensity in the SFG is minimized, and vice versa. The optimal evolution trajectory follows a geodesic path on the Bloch sphere surface, representing the minimal-length evolutionary path from the signal field to the sum-frequency field. This geodesic behavior also indicates that the loss and absorption of the optical field in the crystal medium vanish during SFG. Furthermore, the crystal modulation period for the SFG is determined using the optimal phase-matching function. These results offer valuable insights for the design of compact optical chips and nonlinear optical devices.

DOI
10.1364/josab.581198

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Sum-frequency generation in nonlinear optics is a key technology enabling optical frequency conversion. However, how to most effectively enhance the frequency conversion efficiency in nonlinear crystals remains a fundamental problem; shorter crystal length and lower pump intensity while maintaining high-efficiency frequency conversion are critical for enhancing device integration potential and improving device performance in practical sum-frequency generation (SFG) device design. This study employed the Pontryagin maximum principle to investigate the SFG process. A cost functional was formulated that complies with the minimum constraint during SFG, enabling the derivation of optimal control parameters. A central finding revealed an inverse proportionality between the coupling coefficient and the crystal length: for a given crystal length, the pump intensity in the SFG is minimized, and vice versa. The optimal evolution trajectory follows a geodesic path on the Bloch sphere surface, representing the minimal-length evolutionary path from the signal field to the sum-frequency field. This geodesic behavior also indicates that the loss and absorption of the optical field in the crystal medium vanish during SFG. Furthermore, the crystal modulation period for the SFG is determined using the optimal phase-matching function. These results offer valuable insights for the design of compact optical chips and nonlinear optical devices.
We analyze the dynamics of sum-frequency generation (SFG) and difference-frequency generation (DFG) for broadband input fields. We demonstrate that amplitude modulation of the input fields significantly affects conversion efficiency of SFG and DFG, but frequency modulation does not affect conversion efficiency. Analytic results for SFG and DFG output-field intensities are thereby available for arbitrary fields. Self- and cross-phase modulation can affect the dynamics in the limit of very high field strengths, but these effects are of higher nonlinearity and therefore usually negligible. The quantum efficiency of DFG is shown to be generally higher than SFG, but the energy efficiency is much lower if the difference frequency is significantly less than the input frequencies. Optimal SFG quantum conversion efficiency for such systems approaches that of second-harmonic generation only if the quantum fluence (total number of photons) in the two input beams are equal. Optimal DFG quantum conversion efficiency occurs when the number of photons of frequency ${\mathrm{\ensuremath{\omega}}}_{2}$ is small yet sufficient for significant stimulation. A recently developed method for obtaining efficient SFG for multimode input fields is theoretically analyzed. The method involves using an arrangement with two or more nonlinear mixing crystals with a time-delay line situated between the crystals that delays one of the fundamental fields relative to the other. The efficiency in the second crystal depends on the cross-correlation function of the two fundamental fields upon leaving the first nonlinear crystal. The time-delay method is not effective for multimode DFG.
In this paper, we reveal that some kinds of optical nonlinearities are further enhanced when incoherent light, instead of a laser, is used as a pump light. This idea was confirmed both theoretically and experimentally in the case of sum-frequency generation (SFG) using the optical second nonlinearity. The conversion efficiency of the SFG with incoherent light pumping increased as the bandwidth of the incoherent pump light decreased, finally reaching twice the conversion efficiency of conventional second harmonic generation (SHG) by laser pumping. This method dramatically relaxes the severe requirements of phase matching in the nonlinear optical process. The conversion efficiency became less sensitive to misalignment of the wavelength of pump light and also of device operation temperature when the bandwidth of the incoherent pump light was sufficiently broad, although the improvement of the conversion efficiency had an inverse relationship with the insensitivity to the phase-matching condition. The temperature tuning range was enhanced by more than two orders of magnitude in comparison with the conventional SHG method. As an example of a promising application of this new idea, we performed the generation of quantum entangled photon-pairs using cascaded optical nonlinearities (SFG and the subsequent spontaneous parametric down conversion) in a single periodically poled LiNbO3 waveguide device, in which the incoherent light was used as the pump source for both the parametric processes. We have achieved high fidelity exceeding 99% in quantum-state tomography experiments.
An analytical method is proposed in this paper to achieve complete energy conversion in sum frequency generation based on the Lewis-Riesenfeld invariants theory. In the proposed scheme, a quasi-adiabatic single control parameter model is established, and the value of single control parameter is selected to make the initial eigenstate perfectly converted to the final eigenstate as needed. Corresponding to the nonlinear frequency conversion process, a nonlinear crystal structure is designed by inverse engineering using the optimal control theory. It is robust against perturbations of the coupling coefficient and phase mismatch, including variations in the pump intensity and crystal polarization period, and achieves almost 100% conversion efficiency at any crystal length. Theoretical simulations show that frequency conversion can be achieved in the wavelength range of 2.6 µm-3.6 µm, and the spectral bandwidth of conversion efficiency exceeds 50% and approaches 400 nm when the crystal length L = 1 mm.
Based on the QPM technology and the counter diabatic compensation approach in quantum physics, we propose an optimal scheme that the signal field completely converts into the sum frequency field under the condition of the superadiabaticity. With the geometric rotation plane designed, we subtly reveals the physical mechanism of the systems before and after adding a counter diabatic term. The property of the energy conversion which is analogous to the elastic collision is explained for the addition counter diabatic compensation. With the help of Bloch vectors in atoms physics, we can clearly show the physics process of the complete energy conversion and the physical mechanism for the shortcuts of the superadiabaticity from a new perspective. This new scheme can optimize adiabatic process, shorten the coupling region and nonlinear crystal length, reduce the influence of the dispersion effect, and also implement the robust efficient energy conversion under the condition of relatively weak pump intensity. Not only does this scheme provide a good theoretical basis for the design of nonlinear optical devices, but also its physical thoughts can be widely used in many other fields.
An efficient broadband sum-frequency generation (SFG) technique using the two cascaded optical parametric processes ${\ensuremath{\omega}}_{3}={\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}$ and ${\ensuremath{\omega}}_{4}={\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{3}$ is proposed. The technique uses chirped quasi-phase-matched gratings, which, in the undepleted pump approximation, make SFG analogous to adiabatic population transfer in three-state systems with crossing energies in quantum physics. If the local modulation period first makes the phase match occur for ${\ensuremath{\omega}}_{3}$ and then for ${\ensuremath{\omega}}_{4}$ SFG processes then the energy is converted adiabatically to the ${\ensuremath{\omega}}_{4}$ field. Efficient SFG of the ${\ensuremath{\omega}}_{4}$ field is also possible by the opposite direction of the local modulation sweep; then transient SFG of the ${\ensuremath{\omega}}_{3}$ field is strongly reduced. Most of these features remain valid in the nonlinear regime of depleted pump.
In this thesis, high-efficiency frequency conversion Sum Frequency Generation (SFG) is theoretically analyzed and experimentally demonstrated. Our approach to SFG consists of two input phase conjugated broadband signal and idler beams undergoing SFG. By using the signal waveform as a gating beam, a specific mode may be recognized and separated out, providing a promising method for mode selection. The signal and idler are produced by a monochromatic pump and a broadband signal through Difference Frequency Generation (DFG). To obtain good mode selectivity, high conversion efficiency of the selected mode is required. We study the factors that influence the conversion efficiency of the selected mode, like the length of the waveguide and the power of the gating beam. The highest conversion efficiency of the mode achieved is over 90%. By breaking the correlation between mode and gating beams, we find around 40% conversion efficiency of the mismatched mode.