Application of temporal superresolution for measuring fast single-shot optical waveforms

Abstract

Measurements of ultrafast optical waveforms using phase modulation (so-called time lens) have recently attracted considerable attention [1,2]. These methods allow one to achieve high resolutions up to 10 fs and to measure waveforms with temporal lengths up to 100 ps. However, nonideal lensing, the mismatch between a time lens and dispersions and higher-order dispersions lead to temporal aberrations of measured waveforms. Recently, a method of temporal superresolution based on periodic intensity modulation [3,4] has been proposed that does not require any dispersion and a specific shape of modulation. In this work, we propose the application of this method for measuring single-shot optical waveforms and numerically demonstrate its implementation.We consider the measurement of the digital optical signal "1101" consisting of three Gaussian 1.7 ps pulses (see Fig.1, blue curve) using a photodiode and a conventional real-time oscilloscope with a 30-GHz passband. The waveform intensity spectrum is much broader than the passband of the recording system. In conventional measurements, only a small central portion of the spectrum is transmitted and the signal on the oscilloscope screen will be broadened and smeared. We divide the waveform spectrum into 2N+1 parts with the widths equal to that of the passband of the recording system, and the spectrum is transmitted and measured by parts.First, we obtain the burst of 2N+1 replicas of the input waveform, using , for instance, 2N+1 evenly spaced low reflecting fiber Bragg gratings. Then the burst is periodically modulated by two phase modulators, placed into two arms of a Mach-Zehnder interferometer. The signal that would be recorded with an oscilloscope for a modulation frequency of 40 GHz and modulation indices of 2π is shown in Fig.2. It can be seen in the inset that the initial Gaussian pulses are entirely smeared in each of 2N+1=17 replicas. However, this pattern contains full information on the input waveform. The n-th part of the input waveform spectrum can be found as 2 , , 0where c n is the n-th Fourier coefficient of the modulation function, H(ω) is the transfer function of a photodiode and an oscilloscope, F osc,m (ω) is the Fourier transform of the m-th modulated replica measured by an oscilloscope, and A m is the amplitude of the m-th replica. The waveform spectrum is composed from its restored parts and then inversely Fourier transformed for obtaining the input waveform. The restored waveforms are shown in Fig.1 in absence of noise (red curve) and for a noise level of 1% (green curve). The root mean square deviations from the original waveform are 0.5% and 2.1%, respectively. The maximum temporal length of measured waveforms is determined by the delay time of the replicas and can be increased up to milliseconds.