Creating a Timebase for THz Waveforms

Referring back to Figure 3.2, our overall schematic for the processing of THz waveforms, it is seen that following the acquisition of sample and reference THz waveforms that the next step in the time-domain waveform preparation is the creation of a timebase for the waveform(s). This is an important part of the pre-processing of the acquired time-domain THz waveforms prior to the Fourier analysis for conversion to frequency-domain data. As noted above in the parent section,§3.3 and immediately above in§3.3.1.3, the reference and sample waveforms are discrete time-sampled waveforms having a timebase in the laboratory timeframe. For example, the time axis in Figure 3.5 is in units of milliseconds as, at a 100 Hz repetition rate offset, a total of 10 milliseconds of ‘real-world’ laboratory time is required for the ASOPS technique’s repetitive pump-probe measurements to sweep through the full, approximately 12.5 nanoseconds of delay time. Fundamentally, the generated waveform is one that should have a timebase in the delay timeframe. In the raw data, there may be microseconds or nanoseconds of lab time between digital samples of the waveform, while in delay time, this corresponds to picoseconds and femtoseconds.

The assignment of a proper time axis in the delay timeframe, or more precisely, the conversion of the sampling rate from lab to delay time frame, is of importance for obtaining the correctly scaled frequency spectrum. In performing the numerical FFT analysis of the experimental time-domain waveforms, the FFT software (see§3.4.1) produces a result that is an array of frequency amplitudes, where each position in the array is assumed to correspond to a frequency being a specific fraction of the sampling rate at which the waveform was acquired. While the present waveforms were acquired over several milliseconds (per indi- vidual scan) in our lab-time, they represent an ultrafast process that is actually occurring on a picosecond and sub-picosecond timing level. It is based on this ultrafast timescale that the Fourier results and frequency assignments must be calculated. It can be convenient, then, to convert a waveform such as that shown in 3.5 having time units in milliseconds, over to the proper timescale. This has the benefit of showing a ‘true’ representation of the THz waveform, enabling direct comparisons with such waveforms acquired on tradi- tional delay-line based instruments. Further, this of course enables various time-domain based analyses. For example, one may wish to obtain a direct time-domain measurement of the FID ring-down time or to track down the sources of reflection peaks, by finding their optical pathlength given the delay time to their arrival, the speed of light and the indices of various materials in the beam path.

The conversion of the lab-time sampling rate to one in delay time, and the closely related problem of assigning a delay-time axis to the acquired waveforms in resolved in a straightforward manner. Recall from chapter 2, that the ASOPS effect can be viewed from the perspective of a time-scaling or time dilation effect. The full range of the delay scan is the time period between subsequent pulses of the pump laser. And this

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 1·10−14_{s (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.