A Generalized Mode-Locking Theory for a Nyquist Laser With an Arbitrary Roll-Off Factor PART II: Oscillation Waveforms and Spectral Characteristics (2021)

In this paper (PART II), we present output waveforms and the corresponding spectrum of a periodic Nyquist pulse train with a roll-off factor α emitted from a mode-locked Nyquist laser. In the first part, the relationship between the optical filter amplitudes H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> installed in a Nyquist laser cavity is derived by using the inverse Fourier transformation of filter F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ( ω) at a low frequency edge and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ( ω) at a high frequency edge. We found that the relationship H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> = (4/3) H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> for α = 0 is changed into the relationship H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> = (1/cos( β Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> /2)) H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> for α ≠ 0, where β = π/( 2αω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ), ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> is the zero-crossing frequency and Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> is the modulation frequency. This relationship is important for describing the entire spectral profile of the optical filter installed in the laser cavity. In the latter part, we report how we succeeded in generating a Nyquist pulse train with an arbitrary α value by employing computer simulations with analytically derived optical filters consisting of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ( ω) and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ( ω), H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> , and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> . We found that a Nyquist laser cannot always generate an isolated ideal Nyquist pulse train because there is interference between the wings of adjacent Nyquist pulses. We clarify the differences and similarities as regards filter shape and the corresponding waveform in the time domain of a single Nyquist pulse and a periodic Nyquist pulse train in terms of differences in power P, time-domain distribution τ, spectrum S, and filter shape F. We show that a pure Nyquist pulse train can be obtained with the condition α N > 10, where differences in P, S, and F are less than 1 %, and we present a useful chart showing how to generate a Nyquist pulse train in the GHz region. N is the number of modes in the low or high frequency region. We investigated the time domain orthogonality g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m,n</sub> of the Nyquist pulse train from the laser and found that the orthogonality can be maintained although there is a small interference effect on the wing of the Nyquist pulse.