Supplementary Information for:
Ultrafast and versatile spectroscopy by temporal Fourier transform
Chi Zhang1, Xiaoming Wei1, Michel E. Marhic2, and Kenneth K. Y. Wong1*
1Photonic Systems Research Laboratory, Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong.
2College of Engineering, Swansea University, Singleton Park, Wales SA2 8PP, UK.
I. Difference between the temporal imaging and temporal focusing systems
In the temporal imaging system (Fig. S1(a)), if the time-lens is based on ps-pulse (Tp) pumped FWM, the minimum output pulsewidth (To) is of the order of ps, and the fs signal pulse would not be measured in this system. This is because the fs pulse (Ts) has a large bandwidth, and the pulsewidth of the ps-pulse stretched pump is narrower than that of the fs signal stretched by the same amount of dispersion. Therefore, in the FWM process, the stretched signal is temporally gated by the shorter stretched pump, and its bandwidth cannot fully convert to the idler’s side. After the output dispersion, the compressed pulsewidth is in ps range, as shown in Fig. S1(a).
To quantitatively analyze this configuration in a simplified manner, we assume that the temporal imaging system is based on the 4-f system (the magnification ratio = 1). If the signal pulsewidth Ts < Tp, we can obtain the output pulsewidth To equal to the pump pulsewidth Tp. If the signal pulsewidth Ts > Tp, the output pulsewidth To is equal to the signal pulsewidth Ts. With up to the second-order GVD coefficients, the complete expression for the minimum output pulsewidth can be plotted in Fig. S2(a) and written as follows [S1]:
(S1)
Figure S1 | Configurations of the temporal imaging and temporal focusing systems. (a) In the temporal imaging system, the time-lens is generated by ps-pulse pumped FWM, and we assume that there is no compression/magnification ratio in the 4-f temporal imaging system. This system is not capable of measuring fs pulses. (b) In the temporal focusing system, the time-lens is generated with ps pulse pumped FWM, and this system is capable of measuring the spectrum of the fs pulse. All the signal and pump pulses discussed here are transform-limited.
By contrast, for the temporal focusing system (our PASTA system), the situation is different (Fig. S1(b)). The ideal output pulsewidth To = ΔwsΦf is no longer proportional to the input pulsewidth Ts, but to its spectral width Δws = 4ln2/Ts. From another point of view, since there is no input dispersion, the fs signal pulsewidth (Ts) is much narrower than that of the stretched ps pump, and so the whole spectral width of the fs signal can be fully converted to the idler’s side in the FWM process. Because the fs signal has a large spectral width, the temporal focusing system is capable of measuring the spectrum of the fs pulse, as shown in Fig. S1(b). However, the FWM process still limits the minimum output pulsewidth, which is half the pump pulsewidth (Tp/2), and it restricts the minimum spectral width Δws. With up to the second-order GVD coefficients, the complete expression for the minimum output pulsewidth after the PASTA system can be plotted in Fig. S2(b) and written as follows [S2,S3]:
(S2)
Figure S2 | Comparison of the temporal imaging and temporal focusing systems. (a) In the temporal imaging system, the output pulsewidth (To) is ideally equal to the signal pulsewidth (Ts), but it is limited by the pump pulsewidth (Tp) for short signal pulses. (b) In the temporal focusing system, the output pulsewidth (To) is ideally proportional to the signal spectral width (4ln2/Ts), but its spectral resolution is also limited by the pump pulsewidth (Tp). (In the figures we assumed that these two systems have the identical focal dispersion Φf = 2450 ps2, and the pump pulsewidth Tp = 2 ps.)
With Eqs. (S1) and (S2), we can plot the working range of the temporal imaging and focusing systems in Fig. S2, and illustrate how the output pulsewidth (To) varies with different input pulsewidths (Ts). Here we assume that these two systems have the identical focal dispersion Φf, and that both of them are limited by the pump pulsewidth (Tp); this corresponds to the two horizontal dash-dotted lines shown in Fig. S2. In the temporal imaging system (Fig. S2(a)), the pump pulsewidth limits the minimum signal pulsewidth (Ts > Tp). In the temporal focusing system (Fig. S2(b)), the pump pulsewidth limits the maximum signal pulsewidth (Ts < 8ln2Φf /Tp), or minimum spectral width.
That is why we claimed that PASTA is capable of achieving over 100-nm observation bandwidth, as long as we have the ideal FWM waveguide. The third-order dispersion may become an issue for short pulses in standard types of fibers, such as SSMF or HNLF. However, fibers can be designed and manufactured to exhibit zero third-order dispersion at a particular wavelength, as well as reduced third-order dispersion in a wide range about that wavelength. For example, a fiber with two zero-dispersion (second-order dispersion) wavelengths necessarily has such a region between the two ZDWs. Hence such fibers may need to be used for demanding applications with short pulses. Within its observation bandwidth, the signal under test can have any kind of distribution, which can be a fs short pulse or discrete CW laser lines.
II. Difference between the PASTA and ADFT systems.
Strictly speaking, both the ADFT and PASTA systems are capable of measuring short pulses, with identical accuracy. With up to the second-order GVD coefficients, the complete expression for the minimum output pulsewidth after the ADFT system can be written as follows [S4]:
(S3)
With Eqs. (S2) and (S3), we can obtain the working range of the ADFT and PASTA in Fig. S3. In Eq. (S3), the constant (Tp/2) is replaced by the variable (Ts), which greatly limits the working range of the ADFT scheme with longer input pulsewidths (Fig. S3(a)), while PASTA has better performance in the longer pulse range (Fig. S3(b)).
Figure S3 | Comparison of the temporal stretching (ADFT) and temporal focusing (PASTA) systems. (a) In the temporal stretching system (ADFT), the output pulsewidth is determined by the input pulsewidth (Ts) and the dispersive broadening factor (4ln2Φf /Ts). (b) In the temporal focusing system (PASTA), the output pulsewidth is determined by half the pump pulsewidth (Tp/2) and the dispersive broadening factor (4ln2Φf /Ts). We assumed that these two systems have the identical output dispersion Φf = 2450 ps2, and the pump pulsewidth Tp = 2 ps. The left parts of the two vertical dotted blue lines correspond to the short pulsewidth regions, where these two mechanisms have the similar performance.
In the short input pulsewidth range (e.g. Ts <50 ps, left part of the dotted blue lines in Fig. S3), both ADFT and PASTA achieve similar output pulsewidths. PASTA only adds a time-lens in front of the ADFT, but for the short-pulse measurement, there is no converging feature in the time-lens, and it can be treated as transparent (or like a pinhole). However in practice, though its functionality is transparent, the two-stage FWM process will inevitably add some intensity noise to the output trace. Therefore, the ADFT process usually provides a more stable output spectral trace over the PASTA system, for short-pulse measurement (e.g. Ts < 1 ps). That is why we indicated in the last response letter that, for short-pulse measurements, the PASTA system will be no better than the ADFT technique. This is not because PASTA cannot measure short pulses (or have a pulsewidth limitation), it is simply due to the extra noise introduced within the FWM process.
Supplementary References:
[S1] Kolner, B. H. Space-Time Duality and the Theory of Temporal Imaging. IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[S2] Zhang, C., Chui, P. C. Wong, K. K. Y. Comparison of state-of-art phase modulators and parametric mixers in time-lens applications under different repetition rates, Appl. Opt., 52, 8817-8826 (2013).
[S3] Zhang, C., Wei, X. Wong, K. K. Y. Performance of parametric spectro-temporal analyzer (PASTA), Opt. Express, 21, 32111–32122 (2013).
[S4] Goda, K., Solli, D. R., Tsia, K. K. Jalali, B. Theory of amplified dispersive Fourier transformation. Phys. Rev. A 80, 043821 (2009).
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