This thesis contains two major components of research: numerical simulation of optical-parametric amplification cross correlation of Frequency-Resolved Optical Gating (OPA-XFROG) and numerical simulation of GRENOUILLE and its related issues. Recently, an extremely sensitive technique--OPA-XFROG has been developed. A short pump pulse serves as the gate by parametrically amplifying a short segment of the signal pulse in a nonlinear crystal. High optical parametric gain makes possible the complete measurement of ultraweak, ultrashort light pulses. Unlike interferometric methods, it does not carry prohibitively restrictive requirements, such as perfect mode-matching, perfect spatial coherence, highly stable absolute phase, and a same-spectrum reference pulse. We simulate the OPA-XFROG technique and show that by a proper choice of the nonlinear crystal and the noncollinear mixing geometry it is possible to match the group velocities of the pump, signal, and idler pulses, which permits the use of relatively thick crystals to achieve high gain without measurement distortion. Gain bandwidths of ~100 nm are possible, limited by group velocity dispersion. In the second part of the thesis, we numerically simulate the performance of the ultrasimple ultrashort laser pulse measurement device- GRENOUILLE. While simple in practice, GRENOUILLE has many theoretical subtleties because it involves the second-harmonic generation of relatively tightly focused and broadband pulses. In addition, these processes occur in a thick crystal, in which the phase-matching bandwidth is deliberately made narrow compared to the pulse bandwidth. We developed a model that include all sum-frequency-generation processes, both collinear and noncollinear. We also include dispersion using the Sellmeier equation for the crystal BBO. Working in the frequency domain, we compute the GRENOUILLE trace for practical-and impractical-examples and show that accurate measurements are easily obtained for properly designed devices. For pulses far outside a GRENOUILLE's operating range (on the long side), we numerically deconvolve the GRENOUILLE trace with the response function of GRENOUILLE to improve its spectral resolution. In the last part of the thesis, we simulate the second harmonic generation with tightly focused beams by use of lens. Thus, we are able to explain the `weird' focusing effect that has been a `puzzles' for us in the GRENOUILLE measurement.

Many essential processes and interactions on atomic and molecular scales occur at ultrafast timescales. The ability to measure and manipulate ultrashort pulses hold the key to probing and understanding these key processes that physicists, engineers, chemists and biologists study today. Measuring ultrashort pulses means that we measure both the intensity (which is a function of time) and the phase of the pulse in time. Or alternately we might measure spectrum and spectral phase (in the corresponding Fourier domain). In the early 1990's, the invention of FROG opened up the field of ultrashort measurement with it's ability to measure the complete pulse. Since then, there have been a whole host of pulse measurement techniques that have been invented to measure all sorts of ultrashort pulses. However, no variation of FROG nor any other fs pulse measurement technique, for that matter, has yet been able to completely measure arbitrary ultraweak femtosecond light pulses such as those found in nature. In this thesis, we will explore a couple of highly sensitive methods in a quest to measure ultraweak ultrashort pulses. We explore the use of Spectral Interferometry, a known sensitive technique as one possibility. We find that it has certain drawbacks that make it not necessarily suitable to tackle this problem. But in the course of our quest, we find that this technique is highly suitable for measuring 10s of picosecond-long shaped pulses. We discuss a couple of developments which make SI highly practical to use for such shaped pulse-measurements. We also develop a new technique which is a variation of FROG, based on the non-linearity of Difference Frequency Generation and Optical Parametric Amplification, which can amplify pulses as weak as a few hundred attojoules to be able to spectrally resolve them and measure the full intensity and phase of these pulses. This technique offers great potential to measure generalized ultraweak ultrashort pulses.