Generalized Kramers–Kronig relations and sum rules for moments and powers of degenerate four wave mixing susceptibility

Generalized Kramers–Kronig (K–K) type dispersion relations and sum rules are derived in the static limit for the moments of the degenerate four wave mixing susceptibility. The degenerate nonlinear susceptibility is different from a typical use of the conventional K–K dispersion relations, which assume absence of complex poles of a function in the upper half of complex frequency plane, whereas degenerate susceptibility has simultaneous poles in both half planes. In the derivation of the generalized K–K relations the poles and their order are taken into account by utilization of the theorem of residues. The conventional K–K relations can be used to estimate the real and imaginary parts of the second and higher powers of the susceptibility as the effect of the poles is reduced due to a faster convergence of the dispersion relations. The present theory is directly applicable to higher order susceptibilities and can be used in testing of theoretical models describing the degenerate four wave mixing susceptibility in nonlinear optical and terahertz spectroscopy.

Simplified Near-Degenerate Four-Wave-Mixing Microscopy

Four-wave-mixing microscopy is widely researched in both biology and medicine. In this paper, we present a simplified near-degenerate four-wave-mixing microscopy (SNDFWM). An ultra-steep long-pass filter is utilized to produce an ultra-steep edge on the spectrum of a femtosecond pulse, and a super-sensitive four-wave-mixing (FWM) signal can be generated via an ultra-steep short-pass filter. Compared with the current state-of-the-art FWM microscopy, this SNDFWM microscopy has the advantages of simpler experimental apparatus, lower cost, and easier operation. We demonstrate that this SNDFWM microscopy has high sensitivity and high spatial resolution in both nanowires and biological tissues. We also show that the SNDFWM microscopy can achieve an ultra-sensitive detection based on the electron-resonance effect. This method might find an important application in tracking of nano drugs in vivo.

Degenerate Four-Wave Mixing in Phycoerythrin Dye-Doped Nanoparticles

We have generated a phase-conjugate (PC) wave from nanoparticles with a new microscopic system proposed. The microscope includes a confocal system with a degenerate four-wave mixing (DFWM) system, which plays a major role in generating the phase-conjugate wave to compensate phase distortion in the optical path toward targets. The proposed optical system detects feeble PC wave and imagines 3D particles while improving the inplane contrast resolution of the microscopic image.