Thursday, July 30, 2009
In essence, the project is complete, though there is still a good deal that I can do with my data and even more that I can do with a few more months on this particular setup.
Tuesday, July 28, 2009
Monday, July 27, 2009
This would be really interesting if it worked, as we would be able to build a profile of the resultant beam due to such an obstruction in the way. To look at this theory a bit, I made scans in the y-direction at two different x-positions -- x = max (16.5 mm) and x = 14.0 mm. For the scan at x = max, I used the scan from yesterday as a reference (which is maybe not the best reference), and for the x = 14 mm scan, I took a scan with the "F" in place and a scan with it removed to use as a reference. The only other problem with using such a scan for the reference is that the "F" is held in place with plastic wrap in a frame, and so there is going to be some reflection due to the plastic wrap. The plot for the x = max is shown below.
There seems to possibly be some sort of feature showing up around where the center of the beam is located, though it is not very easy to tell. One big problem ends up being that I do not know where I am scanning along the letter, so I do not know what features to be looking for. I think maybe the best thing to do with this is to be able to take some sort of a ratio of the two signals to see what the relative difference is of the two at each point.
The next plot shows the data I took at x = 14 mm (where the reference was taken right after the scan with the "F" in place).
It is very easy to see that these two scans are extremely similar in almost every way to each other. The only real difference comes in the relative amplitude of the signals, but that is probably due to the plastic wrap used to hold the letter in place and not anything else.
It is an overall very interesting thing to be looking at, though I think the best option is to be able to take a full roster scan of the cross-sectional area of the beam -- both a reference and one with the letter in place. I would like to look into this more, though I will have to see what time permits.
After taking these scans, Antoine wanted to adjust the parabolic mirrors to see if he could correct some of the skewness that I saw in the x-direction scan from earlier in the week. After he adjusted the mirrors, here is the scan that I acquired.
The shape is, in general, much more like what we would expect for a Gaussian beam, but is rather clearly skewed to the left. Also, for the "edges" of this scan, I have already mentioned that we can ignore them due to the lock-in and how it is working. Overall, the shape looks better, and if anything else, then this should suggest that this technique of profiling is indeed rather useful and with some more work might become a very accurate way of profiling such a beam.
In order to see how this scan differs from the scan taken earlier in the week, see the plot below.
The most important thing to notice from this is that changing the parabolic mirrors around and realigning the microscope will result in a change in beam shape to some extent. Adjusting the setup and then doing a scan by hand to test the profile is maybe not the easiest way to do this, but it is definitely a rather straight-forward way to see how the beam looks.
I hope to maybe look into some of these things a bit more in the final week, though there are a lot of things which I may focus on -- the waist size as a function of frequency, the N2 data, the "F" in the beam, etc.
The new y-scan is shown in the plot below. The second plot which directly follows is simply this new scan and the scan from yesterday.
It is pretty obvious that the two scans from today and from yesterday are very similar, with the scan today being a bit more symmetrical. What is interesting to look at is what is happening on the outer edges of this scan from today; after the signal goes to zero. We see that the signal increases a bit and it almost looks as though it could be similar to a diffraction pattern. However, I do not think this is the case.
What happens is that we get the signal by the lock-in being able to read a chopped signal and reference it with when the signal is on and off. When the excitation beam is not incident on the Si then the lock-in is not reading a chopped signal -- it is just reading a stream of signal which ends up just being noise, which is most likely what we are seeing here. It is possible to simply think of it as there being a threshold, after which point the signal does not mean anything. This is what we are seeing.
The dotted black line again represents the excitation beam, the pink components represent the lens and lens arm, which are attached to the translational stage, shown as the grey square. The darker grey squares are the emitter and receiver for the THz signal, the blue ovals are the parabolic mirrors, and the black disc represents a mirror which the excitation beam reflects off. Finally, the grey disc again represents the Si wafer.
After taking an x- and y-lateral scan over the Si wafer, I found some very interesting looking profiles, especially in the x-direction. A plot showing both directional scans is shown below.
Clearly the x-direction is very strange, while the y-direction almost looks like an upside-down parabola. The features in the x-directional scan, I think, help to reinforce the idea that perhaps the incident angle does not have much effect on the profile of the beam and that possibly the alignment of the actual microscope is what should be adjusted in order to correct any sort of skewness. In terms of the y-direction, it would be nice to take a wider range of data; and by this, I simply mean having more data points on the edges of the scan. I hope to do this tomorrow after adjusting the lens and stage.
Overall, it is very interesting to see such an odd profile in the x-direction -- and I think the most interesting thing is that the scan is extremely smooth, as opposed to some more "rigid" scans.
I also should mention that the lock-in was much less noisy in this setup than in the spectroscopy setup. Perhaps this has to do with the THz setup in general, the excitation beam, or something else. Also note that the relative amplitude is much less than what we saw in the spectroscopy setup, where we would see amplitudes of 4 or 5 mV, here we see just over 2 mV.
The black dotted line represents the incident excitation beam and the grey disc this beam is incident on represents the Si wafer, while the grey square represents the translational stage. This is a view from above the setup. The angles of the two beams were not perfectly equivalent, but by "eyeballing" it, the two were very similar (and the exact angle does not matter much, just that the angle is on the other side of the initial angle).
Since the orientation of the stage was not changed, there is nothing to correct for in the beam profiles along the x- and y-directions. The scans for this switched incidence are shown below along with the scan from the initial setup, which I took yesterday.
It is rather clear that these two different scans are almost identical in shape over the x- and y-directions (with the x-direction scan looking almost more similar to the initial incidence scan than the initial vs. switched in the y-direction). The y-direction still gives us a Gaussian and symmetric shape, but with a little skewness maybe in one direction as compared to the initial scan. However, we are most interested in testing the obvious skewness that has been showing up in all of the x-scans as an attempt to see if the incident beam angle was what caused the skewness -- but by looking at the plot of the two x-directional scans, it should be pretty clear that the angle of the incident excitation beam probably does not effect the skewness very much and there is something else that causes this skewness.
One idea (which is pretty simple) is just that there is a misalignment of the parabolic mirrors in the actual microscope design... which is, in theory, why such a profiling technique is useful -- to test the shape of the beam in order to be able to adjust it for the desired shape (probably Gaussian).
Before switching the angle and changing the setup too much, I took a scan along the x-direction and a scan along the y-direction for some kind of a reference after switching the setup (though I already had such scans, I wanted to check that nothing had changed much in the design). Plots of this are shown below.
The first plot is that of the x-direction, while the second plot is of the y-direction. We see very much the exact same features that we have been seeing in such x- and y-direction scans -- the x-direction is rather skewed while the y-direction is rather symmetrical and Gaussian looking. This is not much different from what I expected (if they were different, then that would be a problem with the overall technique).
In order to better compare these scans with the scans done in the earlier weeks on the same setup, I simply plotted the x-direction scans all on one plot and all the y-direction scans on another plot. There were three different scans -- one with the first lens I used which did not focus the beam too well, and then two scans with the new lens in place which gave a tighter beam. These plots are shown below.
The first plot is that of the x-direction, while the second is that of the y-direction. The green data correspond to the scans taken with the first lens and at a larger step size (0.25 mm vs. 0.20 mm). This is most likely the reason why the data from this scan looks so much cleaner in a certain sense -- it was less precise. For the x-direction scans, it is clear that there is the same skewness in each scan (though it is a bit harder to see in the "scan 1"). It is especially clear in the difference between "2nd lens (1)" and "2nd lens (2)", as the only real difference is that these scans were taken on different days, which probably explains the difference in amplitude. All of the same sort of features show up in all of theses scans (and especially those of the 2nd lens). Therefore, it might be safe to assume that this method is in fact accurate to a certain extent.
The y-direction scans need much less explaining -- we see a nice, Gaussian shape in almost all of the scans and the main difference becomes the relative amplitude of the scans, though, again, this can (probably) be attributed to the change in laser power over the course of the day.
In fact, I did not take any scans with the new incidence today, but I did get the stage and lens moved to the proper position. I hope to take these scans tomorrow.
Sunday, July 19, 2009
I do not know the conditions too well under which Antoine took the data (as I will need to talk to him about this), but he simply took a few scans at some of the more interesting points of the THz beam. The plot below shows the spectra for the four points that Antoine took scans at. The plot is labelled, and so it should be clear which spectrum corresponds to which position.
There is clearly some very strange things happening here. I do not understand why there are oscillations for each spectra, and I do not understand how a spectrum could have this shape in general. Before I worry about this too much, I would like to talk with Antoine a bit more about his procedure in taking this data and if they usually see things such as this in spectroscopy measurements under N2 conditions.
For the sake of displaying this data, I have also shown a plot of the temporal trace at each of the points that was scanned. I first show the full time-domain traces, and then for the sake of cleanliness, I chop these pulses to try and "zoom in" on the primary pulse. Both plots are shown below.
The colors for each of the specific positions correspond exactly to those in the spectral plot from above. It is clear from these time-domain traces that we still get the reflections that are so obvious in the time-domain traces done without the N2 environment. It does, however, seem as though there may be a significant decrease in the amount of reflections, though these plots do not rightfully show this.
The following plot is that of the time-domain traces at (0,0) for the N2 environment and under the measurements which I made a few weeks ago in which I took four traces and averaged their signal.
This does not seem to tell too much other than there is a lag between the time of the two pulses (due to the speed of light in the given medium) and there is a clear difference in peak amplitude between the two different environmental conditions. There does, however, seem to be a similar amount of reflections/noise after the main pulses for both scans, which might suggest the N2 did not do much to decrease said reflections.
Finally, it is interesting to look at a comparison between the two spectra -- one at (0,0) in N2 and one at (0,0) in the typical environment. Such a plot is shown below, using the same data as the time-domain trace comparison above.
We still see this oscillatory effect in the N2-environment spectrum, while we see nothing of the sort when not using N2. Also, it is clear that there is a much greater percentage of the spectrum which is transmitted through the wafer, as the amplitude of the N2-environment spectrum is greater than that of the other spectrum. Again, I am not very certain why we see such effects using the N2 as opposed to not, but I would like to investigate this a good deal more.
In order to attain a better understanding of these absorptions and anomalies in the spectral plots, I think I first need to be able to truncate the data from the time-domain pulses to hopefully result in a better, smoother-looking spectra. From here I should be able to better determine waist-size dependence on frequency.
1.) Move everything from the spectroscopy setup to the ellipsometry setup and try to recreate these same results.
2.) Continue trying to understand the specific physical phenomena which occur using my current setup (spectroscopy). This includes mainly (but is not limited to) the skewness in the x-direction of the cross-section of the THz beam and the beam waist size as a function of frequency.
I might feel more comfortable working more on the second choice, though this is yet to be decided.
Thursday, July 16, 2009
Wednesday, July 15, 2009
We can see that the shape is indeed rather symmetrical, just with the other scan in the y-direction. There is a fair amount of skewness, as well. The general shape is Gaussian, but it seems to be too "rigid" to be a proper distribution. The whole idea from this scan is that the profile is rather different than what we would expect for a perfect Gaussian distribution.
The problem with doing a series of scans like this over multiple strips of the Si wafer is that we are able to find some rather interesting behavior, but do not have a very good idea what is happening in regions other than those which are scanned. This can be solved in a variety of ways, the easiest being switching out the current translational stage with an automatic one and writing a simple program to profile the entire beam. The problem with this is time and lack of equipment. I have about two weeks remaining, there is not a translational stage laying around that I could use, and there are perhaps more interesting things for me to be looking at.
This being said, I would like to comment on the noise in the signal that I am reading off the lock-in. In case I have not mentioned this before, I take these data manually, and to try and deal with the noise I take and average the highest value and the lowest value over the duration of a few moments. The signal on the lock-in happens to oscillate back and forth, and these maxima and minima are approximately what I average to get the value that I use to plot.
Antoine and I tried to cut-down on this noise later in the day by changing the chopping frequeny, but the best frequency we found was about 283 Hz (frequencies of about 250 Hz and 300 Hz gave a lot of noise due to the surronding environment). In an attempt to see if this averaging technique changes much the final data, I plotted all three sets of data over position (which includes the maximium, average, and minimum). All three series of data seem to correspond well with each other (which simply means that the difference between maximum and miniumum remains relatively consistent, regardless of the position on the wafer -- thus even further suggesting that it is in fact noise). A plot of this is shown below (Note, this is from the same scan that I am describing above, but I see this effect in every scan that I plot maximum, minimum, and average on).
We thought of some ways to reduce this noise, but could not figure anything out right away. I think the easiest thing would be to connect an oscilloscope directly to the lock-in that is able to look at the signal over time and average the signal even more than what the lock-in already is. From this we could hopefully attain a more accurate average value. This idea would not work, however, since we do not have such an oscilloscope around the lab. Also, because of the consistency with the change in signal due to noise, we might as well ignore this for now.
From here I would like to further look into the differences in signal over position and also to continue trying to learn about the waist of the beam as a function of wavelength.
Friday, July 10, 2009
Thursday, July 9, 2009
This is a plot of the (0,0) position on the Si wafer. Like I said in a recent post, there are three distinct pulses -- the initial pulse which has the greatest amplitude which is then followed by a pulse of slightly less amplitude and then yet another peak of slightly less amplitude. If we look at the horizontal scale, which is measured in picoseconds, we can see that the difference between peaks is approximately 12 ps. We also notice that the three peaks are equally spaced with this same increment.
The math for this calculation is pretty straight-forward. Basically, we have the Si wafer of width l and refractive index n. Most of the signal transmits directly through the wafer, and is seen in the primary pulse in the above scan. Then some of the signal reflects off the boundary between the back of the Si disc (call it b), then reflects onto the first boundary which it has already passed through between the front of the Si and the air (call this a), and finally back through b. This gives us that second pulse. The third pulse is explained in precisely the same fashion -- signal transmits through a, reflects off b, reflects off a, reflects off b, reflects off a, and is then transmitted through b. Since the thickness of the wafer determines the path length of each piece of the pulse, they each show up at a different position on the time scale.
I understand the basic physics behind this effect, but I am interested in determining a method to effectively remove it from the data.
I also spent some time in changing the lens used to focus the excitation beam. We found a lens which had a longer focal length so as to get a tighter focus of the excitation beam on the Si wafer.
After doing this, I took more lateral scans across the Si wafer to determine if the beam shape is still about the same as before. The data that I found are shown in the plots below. First is that of the x- and y-direction on the same set of axis.
The scan traced with the blue line and markers is of the x-direction, while the red line and markers display the scan for the y-direction. We can see from the above plot that the beam profile for the x-direction seems to be even more distorted than what we saw when using the other lens. The reason for this, as Antoine explained, might be due to a low-pass filter effect in which the size of the excitation beam effectively changes the profile of the THz beam. Basically, since the excitation beam is so much tighter and we are creating more accurate measurements than with the wider beam from before, we may expect some of this sort of over-distortion to come about. We also see the same sort of skewness in this scan as the x-scan that was done with the other lens.
The y-scan shows the same sort of Gaussian distribution from before. It is still rather symmetric and looks relatively normal. It is also interesting to note that the peak amplitudes for the two scans do not match up perfectly. I am not really sure what this is due to, since I took both scans within about 30 - 60 minutes of each other.
The distortion in the x-direction and the normal, Gaussian shape in the y-direction suggest that perhaps the x-incidence excitation beam is distorted in some way and thus we see such strange profiles for the x-scans.
To further investigate the matter, I took another scan in the x-direction but at about -HM of the y-direction. This plot is shown below.
It is clear that we see the same sort of distortion in this scan... and skewness. Again, I feel as though the reason for this might be the incidence of the x-beam on the wafer. It would be interesting from here to take even more lateral scans and try to actually build a beam profile from this. I might look into taking what one might consider to be circular contours about the xy-plane to see if I can create a better 3D image, though there may be more interesting things to do.
Tuesday, July 7, 2009
Temporal signal with spectrum side-by-side
Spectra of all the data on the same set of axes
Spectra of positive and negative HM of both the x- and y-directions
Ratio of two spectra
I found some interesting things, but did not spend too much time on quantifying the data. Following are a series of images from each of the four characteristic plots that I mention above.
First is a plot which shows the temporal pulse and the spectrum of this pulse in side-by-side plot windows.
This happens to be the average over four sets of data at the maximum, or most intense, part of the THz beam. In the temporal pulse to the left, it is noticeable that there is the initial peak of the pulse and then shortly thereafter and spaced equally apart there are two other peaks. We think this is due to a Fabry-Perot effect in which the THz signal reflects within the Si before transmitting. I would like to write a script which removes this effect, as it has an influence on the spectrum.
The plot to the right shows the spectrum of this same temporal pulse. The green is the actual spectrum while the blue is the same spectrum after being smoothed out.
Following is a plot which shows the spectra of at each point. This includes the maximum position and the positions for the HM of both directions.
The legend shows the coordinate for each of these spectral scans. It is interesting that the +/- scans at the FWHM in the x-direction do not match up better. This is a little bit more clear in the following plot, which compares the +HM to the -HM of the x-direction.
There is clearly some great variation in these two spectra. Recall that the beam did appear to be skewed in the x-direction, and perhaps this is part of the explanation. We see some different things in the y-direction, which, contrary to the x-direction, seemed to be more symmetric. This plot is shown below.
These spectra seem to be much more symmetric, though there is that sharp valley for the spectra of the +HM direction (recall that I have labelled it ++HM since the other value of +HM seemed to be off). I have not yet taken the time to try and explain this sharp drop.
In order to look at the spectra relative to the reference spectra, I divided the spectra at each position into the spectra for (0,0). A sample plot of this is shown below.
This happens to be the ratio of the (+HM,0) position. This is plotted on a double y-axis plot to show the ratio in green (corresponding to the right vertical axis) as well as the spectrum of the given position (in red) and the spectrum of the maximum position (in blue). Both of the spectra correspond to the left vertical axis.
The idea of taking this ratio is to determine the beam waist size as a function of frequency, though I need to read into this more.
A few more things to look at are possibly taking an even better spatial profile of the beam since there is such a difference in the x-position. This will also allow me to find a better estimate of the FWHM in both x- and y-directions. I also need to look at how our excitation beam may be destroying the Si and thus determine if this effect is negligible for our purposes or not. Also, there is the idea of removing most of the moisture content of the air via replacing it with gaseous nitrogen.
Monday, July 6, 2009
I changed the time constant from 1 second per point (as we used earlier this week) to 100 ms. This changed the length of the scan to take about 9 min instead of the 90 min from earlier in the week. I took four scans at each of the above positions using the positions of FWHM that Antoine had determined. The only problem that I ran into was the position for the (0,+HM) gave scans a lot different from the others and the position was about 0.5 mm different from what I estimated on my initial data. Because of this, I took one more scan at the point I had expected the FWHM to be at. I labelled this point as:
For the first five positions listed above, I took data in a cyclic permutation, but for the point (0,++HM), I just took the four scans all right after each other.
During these scans, I was able to update my lab notebook and I made a note to look at how a Fabry-Perot effect might alter this data.
I tried to learn how to take the ratio with MATLAB but was unsuccessful. I talked to Guilhem about this and he gave me some commands to use. I plan on trying this out either this weekend or sometime next week. I should also have plots of the data early next week.
Unfortunately, I did not have time to take more data today. Instead I worked with MATLAB and importing data from the scans. To do this, I used specific functions that Guilhem wrote to handle such data. With said functions, it is possible to import and build the signal we initially took (temporal). Given a series of indexed data like what I mentioned above, this function will average those signals into one. The other important function was to take this temporal signal and create the spectrum.
I was able to make a few plots with this, one of which shows the spectra of each scan. This plot is shown below.
Notice that the red spectrum is that at the maximum position (0,0), and is greatest in amplitude, just as expected. We would expect the spectra colored in orange and green to be more symmetic, as they are on the positive and negative FWHM of the x-direction and so we would expect them to be very similar. Also, in this scan I took only one measurement in the y-direction. This is because the intensity plot showed the y-direction to be more symmetric than the x-direction (see plot from yesterday).
We can see from this plot that there are distinct frequencies at which there is no signal. This may be caused by absorption via water in the air. We might try to reduce this by putting the microscope in an air-tight container and filling it with nitrogen to remove the water.
From this point, I want to look at the relative ratio of signal at each point on the wafer. To do this, I need some sort of a reference spectrum to compare the other spectra to. The way to do this will be to use the (0,0) maximum point as the reference. I want to divide the spectra from each other position into this reference spectrum, and I should look at this in the next few days.
Also, I have added the compilation of the temporal scans, which is shown below. It is not very easy to see each pulse, but this plot is just to show the relative intensity of each position.
So I will hopefully be able to find the ratio of these two signals tomorrow and also take some more data which I can average out to attain more precise spectra.
Wednesday, July 1, 2009
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.
I began the day (Tuesday) with trying to center the excitation beam on the Si wafer. I did this by first roughly guessing where the maximum x- and y- displacements were and then performing two manual scans: one along the horizontal with the vertical displacement constant at maximum and then the same thing for the vertical scan (horizontal position constant at about maximum value).
These initial scans I did in increments of 0.5 mm and scanned about 7 mm in the x-direction and about 5 mm in the y-direction. Also, note that I have set x to be horizontal and y to be vertical. The idea of these scans was pretty simple: move the excitation beam across the wafer and record the change in THz signal from the lock-in. The greater the change, the more effective this local mirror was.
Plotting the points from these scans showed some pretty nice looking curves for such quick/crude measurements. Both curves were Gaussian in shape, with some minor anomalies.
Note: These initial measurements were done with an aperture which did two things: limited the amount of THz signal and made it possible to see a Fabry-Perot effect between the aperture and the Si wafer. This is why it was removed in later measurements.
Note: The signal tended to be rather noisy (the signal would oscillate in upwards of 0.1 mV at each position, with the relative maximum being about 4 – 5 mV). To reduce the effective noise, I looked for the signal to ‘oscillate’ and took the maximum and minimum values, later to be averaged together.
I then decided to take more precise measurements of this signal and so I changed the step size to 0.25 mm. I first did this with the x – position and plotted it. I found it to be slightly skewed, but still Gaussian. After this measurement, I removed the aperture and took scans in both the x- and y- directions. I found some very nice looking Gaussian curves for both of these scans, with the x- still being slightly skewed.
These scans ended up being pretty convincing that the THz beam is indeed Gaussian in distribution. I approximated the FWHM of the x- and y- directions to be 2.5 mm and 1.5 mm, respectively. I found the maximum signal difference to be 4.548 mV. The integration constant was 1 s, with 24 dB and 10 mV sensitivity, which were all set on the lock-in.
I also need to check if the excitation beam is stretched too much in the horizontal direction because of its angle of incidence on the wafer. I also need to see if the translation on the stage corresponds to the same translation on the wafer (i.e. if moving the stage 2 mm moves the beam 2 mm).