3.3 THz Time-Domain Waveform Preparation
3.3.2 Creating a Timebase for THz Waveforms
delay time is explored at the rate of the repetition rate offset; that is, it is scanned in an amount of time equal to the inverse of the repetition rate, so 100 Hz offset yields 10 ms scans for example. This can be expressed as a simple scale factor, relating a lab-time period to a ‘true’ delay time period:
S= fr
∆f (3.45)
wherefris the repetition rate of the pump laser (typically∼80 MHz herein),∆f the offset in repetition rates between the pump and probe lasers (typically 100 Hz herein), andSis the ratio of lab time duration to delay time duration. For the typical frequencies noted here,S≈8·105. That is, a∼10 ms duration (in lab time) scan reduces by a factor ofS=8·105to 12.5 ns of delay, as we expect.
Similarly,Sscales the digitizer sampling rate:
fASOPS=fADC∗S (3.46)
where fADCis the rate (in lab time) at which the digitizer acquires samples of the time-domain waveform, andfASOPSis the,S-scaled acquisition rate in delay time. For the 125 MSa/s digitizer rate employed, as noted above in§3.3.1.2, for the water vapor scans presented herein, and the typical S-factor value, we find that fASOPS=114samples per second! That is, the effective delay-time sampling rate for the water vapor scans presented here is at∼100 TSa/s (terasamples per second). This is the sampling rate that is used to provide the frequency axis of the FFT results below.
More immediately, for our present purpose of displaying the time-domain waveform in delay time units, each subsequent data point is treated as arriving an amount of time later equal to the sampling time, which is merely the inverse of the calculated∼100 TSa/s rate, or a value of 1·10−14s (10 fs) between subsequent points in the time-domain waveform. In equation form,
τASOPS=τADC
S (3.47)
whereτASOPS is the time between adjacent sampled points in the delay-time waveform and is simply the inverse of fASOPS, andτADCis the time between subsequent samples in the lab-time waveform and similarly, is the inverse offADC. These simple timing relationships, used every time a THz waveform is processed, have been coded into a pair of MATLAB functions, ‘asopsRate.m’ and ‘timebase.m’, included in the appendix,
§A.2 and§A.3. The first function, ‘asopsRate.m’, implements eqs. (3.45) and (3.46) to simply calculate the delay-time sampling rate, typically used to assign the frequency axis to FFT output, as discussed further in
§3.4.1. Another use of the scaled sampling is for the generation of a delay-time axis for an originally lab-
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Time(ps)
Signal (arb. units)
2 Torr H2O Reference
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Figure 3.6:The time-domain THz waveform for both of water vapor and a reference scan. The main view is zoomed-in to the period at and just after the main THz pulse. The inset shows a further-zoomed period, just after the main pulse, so as to clearly show the continued THz-frequency ringing from the water vapor.
time waveform. This is achieved with the ‘timebase.m’ function. It accepts the scaled rate of ‘asopsRate.m’
and calculates a new time array for a waveform, to replace the lab-time, hardware-generated axis from the original acquisition. These two functions have been used to perform the timebase conversion for a water vapor time-domain scan, like that in Figure 3.5 (this scan from the 2-detector arrangement, otherwise similar).
The delay-time waveform is shown in Figure 3.6; it should be noted that this is a zoomed-in view span- ning 1900 ps in the main figure window and 12 ps in the inset window. The main zoomed in view is selected to illustrate the substantial difference in the time-domain signal between the reference and water vapor wave- forms. A problem in trying to plot this many points is the aliasing and undersampling and otherwise poor translation of almost 200,000 data points into small figure—this contributes to making the data appear much noisier than it really is. For this reason, the inset view has been provided, spanning only 12ps and show- ing that there is clearly a water FID well above the background signal level. This FID signal, based on the converted time axis, can clearly be seen to span at least∼1 ns in time before falling down to noise levels.
The MATLAB script used to generate this figure is also included in the appendix,§3.6. It should further be noted that the overall signal level of the sample waveform is much reduced from that of the reference; be- tween this diminished time-domain amplitude and the time-domain ringing long after the initial THz pulse,
we can be confident of seeing clean, distinct spectral features following the Fourier analysis discussed in§3.4 concerning the second part of the overall analysis process, that is, of the discrete Fourier transform of the data.
Prior to the Fourier analysis, in the usual data analysis, there are two remaining aspects to the waveform preparation in this first part of the overall process. In particular the proper portion of the waveform should be selected for further analysis and further, the waveform may benefit from ’windowing’, that is, apodization, to prevent frequency domain ringing. While these two forms of processing are of course done prior to FFT analysis, they are motivated primarily by their effect upon the FFT results, so as noted above, we will first discuss the FFT processing.