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Journal of the Optical Society of America B

Cross-phase modulation caused by cascading of third-order processes

Solomon M. Saltiel, Kaloian Koynov, Krassimir Kirov, K. Petrova

Published Feb 1, 1999• 3 citations
Publisher

Abstract

The evolution of the phase of a weak signal wave (ω2) copropagating with a strong gate wave (ω1) in a centrosymmetric medium is studied analytically and numerically for the case when the two waves are involved in a nearly phase-matched third-order process that generates a third wave (ω3=2ω1+ω2). It is shown that the signal wave collects, in addition to the phase shift that is due to the ω2=ω1+ω2-ω1 process, an extra nonlinear phase shift caused by cascading of two third-order nonlinear processes, ω3=2ω1+ω2 and ω2=ω3-2ω1. The attractive feature of this new type of cross-phase modulation is the fact that one can control its magnitude and sign by changing the wave-vector mismatch of the sum-frequency mixing process and the gate intensity.

DOI
10.1364/josab.16.000262

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The evolution of the phase of a weak signal wave (ω2) copropagating with a strong gate wave (ω1) in a centrosymmetric medium is studied analytically and numerically for the case when the two waves are involved in a nearly phase-matched third-order process that generates a third wave (ω3=2ω1+ω2). It is shown that the signal wave collects, in addition to the phase shift that is due to the ω2=ω1+ω2-ω1 process, an extra nonlinear phase shift caused by cascading of two third-order nonlinear processes, ω3=2ω1+ω2 and ω2=ω3-2ω1. The attractive feature of this new type of cross-phase modulation is the fact that one can control its magnitude and sign by changing the wave-vector mismatch of the sum-frequency mixing process and the gate intensity.
Both static and dynamical analyses of quasi-phase-matched second-harmonic generation by backward propagating interaction, where the second-harmonic wave is generated in reflection, are presented. It is shown that bistability appears in the generated second-harmonic power if the amount of phase mismatch is suitably chosen. The bistability also appears in the output fundamental wave, since the cascaded second-order nonlinear process gives rise to the effective self-modulation of the fundamental wave, which can be applied to all-optical signal processing. The output second-harmonic and fundamental powers can be switched between two bistable states by controlling the intensity or phase of the input fundamental wave.>
A comprehensive analytical approach to scalar and vectorial interaction in cascaded quadratic nonlinearities is presented. Exact formulas for the intensity as well as the phase modulation in dependence on all conceivable control parameters are derived, provided that group-velocity dispersion, diffraction, and pulse walk off can be neglected. The first two constraints are uncritical as far as channel waveguides and thin bulk crystals are concerned. Particular emphasis is paid to a detailed study of the phase evolution in order to identify optimum conditions for an almost homogeneous phase shift across an entire pulse which is in contrast to the generation of a large chirp in cubic nonlinear materials. Moreover, the general results are simplified to relevant limiting cases where simple analytical formulas emerge for the amplitude and phase modulation. The analytical results are used for a classification of all-optical effects based on cascading. Optimum conditions for picosecond modulation processes are discussed. \textcopyright{} 1996 The American Physical Society.
Cascaded nonlinear phase shifts via second-harmonic generation are theoretically studied for quasi-phase-matched (QPM) structures. Both the effective phase mismatch and the structure itself contribute to the resultant nonlinear phase shift. Depending on the sign of the effective phase mismatch, structural contribution may enhance or depress the nonlinear phase shift. The fundamental wave will experience a nonlinear phase shift even if the effective phase matching is satisfied. By properly designing the effective phase mismatch, QPM structures with large material-based phase mismatch may provide large nonlinear phase shifts before saturation.
The resurgence of interest in cascaded second order nonlinearities[1,2] has resulted in several schemes for applying cascading in second-harmonic generating (SHG) crystals to optical transistor-like devices [3-7], Such devices allow a weak signal beam to impose a large amplitude modulation on a strong “pump” beam, through mixing in a χ(2) crystal. However, all of the devices reported to date require the pump and signal beams to be temporally coherent, as the process is strongly dependent on the phase between these two beams. There have been some suggestions[6,7] that type II phasematched SHG may remove phase sensitivity, but the devices proposed still require that the signal must be in phase with approximately half of the pump intensity. In this letter, we describe how one may properly exploit type II phase matching to produce a truly phase-insensitive all-optical transistor (AOT) based on second order nonlinearities.
We calculate the spectrum of third-harmonic generation (THG) for multimode fundamental input fields. We consider input fields from (1) a laser with gain on several cavity modes, (2) pulses with short temporal duration, and (3) frequency-modulated input fields. The effects of phase modulation in the nonlinear medium on the spectrum of the THG field are considered. Using the estimates by Miles and Harris [IEEE J. Quantum Electron. QE-9, 470 (1973)] for the nonlinear susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-3\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) in alkali-metal vapors, we find that phase modulation can profoundly affect the efficiency and the spectrum of the THG. Generally, the self-phase modulation of the fundamental will tend to reduce the phase matching of an intense fundamental field and the third harmonic and thereby lower the conversion efficiency. New frequencies not contained in the set of frequencies ${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{j}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{k}}$, where {${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$} are frequencies in the fundamental laser spectrum, are created even at relatively low conversion due to phase modulation. Effects of cross phase modulation are also studied.