Theory of FM Mode-Locking of a Laser as an Arbitrary Optical Function Generator

Abstract

We describe theoretically a new method for the generation of various pulses from an FM mode-locked laser, where we install a specific optical filter <inline-formula> <tex-math notation="LaTeX">$F_{F}(\omega )$ </tex-math></inline-formula> in the laser cavity, which is characterized by <inline-formula> <tex-math notation="LaTeX">$A(\omega )$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$A(\omega +n\Omega _{m})$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$n= - \infty \sim +\infty $ </tex-math></inline-formula>. Here <inline-formula> <tex-math notation="LaTeX">$A(\omega )$ </tex-math></inline-formula> is the Fourier transformed spectrum of the output pulse <inline-formula> <tex-math notation="LaTeX">$a(t)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Omega _{m}$ </tex-math></inline-formula> is the phase modulation frequency. It is important to note that <inline-formula> <tex-math notation="LaTeX">$F_{F}(\omega )$ </tex-math></inline-formula> is a complex filter consisting of real and imaginary parts. We also install a Nyquist filter with a roll-off factor <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> for the bandwidth limitation of <inline-formula> <tex-math notation="LaTeX">$F_{F}(\omega )$ </tex-math></inline-formula>, which is indispensable if we are to form a pulse with FM mode-locking. With AM mode-locking, we derived <inline-formula> <tex-math notation="LaTeX">$F_{A}(\omega )$ </tex-math></inline-formula>, which consists of only a real value in most cases, whereas phase (or frequency) modulation accompanies a phase rotation in each longitudinal mode. We analytically and numerically show that Gauss, secant hyperbolic (sech), parabolic, and triangular pulses are generated in a similar way to the AM locking that we have already reported. We also show that a chirp-free pulse can be generated with the present method, which would not be possible with conventional FM mode-locking. In addition, and more importantly, we can generate a waveform that has constant amplitude and vertical amplitude parts in the time domain such as a rectangular pulse, a triangular intensity pulse, and a parabolic intensity pulse whose waveform looks like a &#x201C;similariton&#x201D; familiar in nonlinear fiber optics. This is because phase modulation does not induce an amplitude change in the time domain, while AM modulation directly changes the amplitude. The generated rectangular pulse has ringing at rise and fall positions due to the Gibbs phenomenon. We also describe a difficulty for FM mode-locking as regards the generation of Nyquist pulses with <inline-formula> <tex-math notation="LaTeX">$0 \le \alpha \lesssim 0.2$ </tex-math></inline-formula> although we can generate a Nyquist pulse with <inline-formula> <tex-math notation="LaTeX">$0.2\lesssim \alpha \le 1$ </tex-math></inline-formula>. This is because the wing of the output pulse for <inline-formula> <tex-math notation="LaTeX">$0 &lt; \alpha &lt; 0.2$ </tex-math></inline-formula> is not sufficiently confined within the duration (<inline-formula> <tex-math notation="LaTeX">$\pi /\Omega _{m})$ </tex-math></inline-formula> where the frequency chirping occurs on the same slope.