Theory of AM Mode-Locking of a Laser as an Arbitrary Optical Function Generator
Masataka Nakazawa, Toshihiko Hirooka
Abstract
We present theoretically an AM mode-locked laser that can generate various kinds of optical pulses. By employing a non-perturbative master equation in the frequency domain, we show that we can design an arbitrary output pulse waveform, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a(t)$ </tex-math></inline-formula> , output from a laser with a specific optical filter, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{A}(\omega)$ </tex-math></inline-formula> , characterized by a Fourier transformed spectral profile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A(\omega)$ </tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a(t)$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A(\omega +\Omega _{m})$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A(\omega -\Omega _{m})$ </tex-math></inline-formula> . Here, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Omega _{m}$ </tex-math></inline-formula> is the AM modulation frequency. Although the optical filter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{A}(\omega)$ </tex-math></inline-formula> generally has a complex frequency response, most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{A}(\omega)$ </tex-math></inline-formula> filters are characterized by real values as long as the mode-locked pulse waveform is symmetric in the time domain. However, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{A}(\omega)$ </tex-math></inline-formula> becomes spectrally complex when our aim is to generate an asymmetrically mode-locked waveform, for example a single-sided exponential pulse. The actual <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{A}(\omega)$ </tex-math></inline-formula> can be designed by using, for example, a liquid crystal on silicon (LCoS) optical filter, which can simultaneously control the amplitude and the phase of the input signal. A sech pulse (soliton) has already been generated based on the nonlinear Schrödinger equation by using Kerr nonlinearity in a fiber, but we show in this paper that the pulse can be generated very precisely even without nonlinearity. Since the present method enables us to generate triangular, double-sided exponential pulses as well as Gaussian, sech, parabolic, and even Nyquist pulses in the amplitude expression, we may be able to use AM mode-locked lasers as optical function generators.