Wednesday, July 1, 2009

MATLAB and Gaussian Curves

I spent the better part of the day (Wednesday) learning how to use MATLAB. The main source that I used to do this was “MATLAB Guide”, by Desmond J. Higham and Nicholas J. Higham. I made it through the first chapter or so and did some interesting examples including Mandelbrot set, Fibonacci numbers, and simple 2D plots.

I was then able to use MATLAB to input my data from yesterday and plot the two curves on the same set of axes (though it would have been nice to tell MATLAB to put them on orthogonal axes).

Note: I corrected for the x- and y- positions by setting the point of maximum intensity at (0,0).

That data is shown in the image below:

The red shows the data from the x-scan, while the blue corresponds to the y-scan. Notice the final data point at the origin for the y-scan. This is simply because I put the data from the two scans into the same matrix and there were more points for the x-scan than the y, and so the these points were all plotted at (0,0).

Also, below is a plot that Antoine constructed with the same data by using an iterative procedure in MATLAB:

[I apologize for the poor picture quality, but I had to use Paint since I saved these as PDF at lab and Blogger couldn't insert a PDF].

Regardless of the quality, it should be clear that the beam is Gaussian in shape (at least via this method).

I would like to make some more scans to better build up images such as those above, which would probably include scans along y = x and y = -x.

Aside from learning to use MATLAB today, we took a series of scans at specific positions of the excitation beam. We took a scan at maximum (0,0), at maximum x and the lower portion of the HM of y, and then at maximum y with both the upper and lower portions of the HM of x. These were full spectral scans, and each took about 1.5 hours to complete.

Note: There was a major shift in maximum signal change from yesterday to today. The maximum went from about 4.5 mV yesterday to about 7.5 mV today. This may be due to a few things: more power out of the laser OR the excitation beam is locally destroying the Si OR some other thing yet to be determined.

Antoine was going to take one final scan today with the Si wafer slightly shifted so that if the effective change in amplitude was due to a local defect then we should be able to see it right away. For tomorrow, I intend to work some more on learning MATLAB and possibly changing the incident excitation beam on the Si to try and understand the skewness of the x-position (though I think it may have to do with a slight tilt in the wafer (i.e. not perfectly orthogonal to the incident THz beam)). Also, if I have time then I may try to take some more data, as I had mentioned, with the scans being along y = x and y = -x.


  1. Those are really nice looking profiles for the Gausian THz beam. I actually prefer the 2 scans of x and y, as they are more quantitative than the color plot.

    I completely agree that the skew in the x-direction is likely from the elliptical nature of the optical excitation beam that hits the Si from an angle other than normal incidence.

    I have doubt that the laser is destroying the Si. If the excitation beam is bigger than maybe 10 um diameter (probably even less), it won't have enough intensity to damage the semiconductor (probably not enough pulse energy to hurt the Si either). If the beam is smaller, it might ablate some material away, but it would be over an extremely small area. If that caused a difference in the transmitted THz when the excitation beam was on, then it would probably also cause the transmission to be different than that observed for the original (i.e., undamaged Si) reference experiment. I would guess more likely that the optical power had changed from one day to the next, some alignment had changed in the THz set-up, or something similar. Do you feel as if the same initial tweaking for optimization of the THz signals was done before both experimental days? I ask, because, for instance, just a tiny difference in the position of the probe beam on the receiver photoconductive gate can easily lead to the difference in signal amplitude that you describe.

    I don't think you ever mentioned if the last scan by Antoine with the Si wafer shifted laterally gave a result consistent with the damage theory. I would expect he did not see a change in THz signal amplitude, but I am curious as to this result.

    I would suggest doing a reference scan first thing every day, after optimization of the THz signal. We have found that even during the course of the day, periodically re-checking the resistance of the photoconductive elements (and slightly tweaking the pump and probe beams on the photoconductive gaps to minimize the resistance of the elements) is a beneficial procedure that will yield better results by correcting for system drift.

  2. As I may have already mentioned (either in a later post or via email to John), I think the skewness is in fact consistent with the theory of the elongation of the excitation beam on the Si wafer -- and this is to be investigated in little time.

    In terms of the shift in maximum signal, I also find it esy to believe that the fs laser has changed its power from day-to-day or even through the course of one day. However, based on what Antoine found when he moved the wafer was a drastic drop in the signal difference (from about 7.5 mV at maximum to about 4.5 mV).

    Now, I think that there may be another explanation for said behavior other than a destruction of Si at local points on the wafer, as I had earlier mentioned. I think maybe the easiest explanation for what Antoine saw was that perhaps in moving the wafer, we effectively changed the angle of the plane of the Si so as to either reflect or transmit the excitation beam in a different way. This would therefore result in a possible change in signal.

    I think that the best way to avoid all of this is to simply take all of the scans conseculatively and to take multiple scans in a cyclic manner so as to avoid this effect and compensate for variations in fs laser power throughout the day.

    This works for both the profiling of the beam (which does not involve me taking a time-domain trace), and in temporal scans which I use to look at the frequency dependence of the waist of the beam. Perhaps doing a roster scan right away and determining the profile of the beam is the best thing, and then from there simply knowing the positions of maxima, FWHM, etc. can be used to take the temporal scans at a later time.

    In saying all of this, I would like to maybe take one day and complete a comprehensive roster scan with a given incidence of the excitation beam on the Si wafer. The effect of the changing signal can be looked at as well, but for now my main concern is to look at the overall profile of the beam.