Set the cooler to 5 deg C initally (Temperature outside is about 13 deg C).
Set exposure time to 0.1s.
1. Data Taking & Analysis
Firstly, we centred the 3 peaks in the frame by changing the grating angle - the setting on the micrometer was 3mm.
Focus
We then changed the distance between the camera and the lens to alter the focus. We intend to take a range of frames at different focuses such that we can plot the standard deviation (stdev) of the width of the spectral line as a function of this distance, in order to find the best focus distance.
We started the focus distance at 21mm, where the image was clearly out of focus. We decreased in increments of 0.1mm each time until we reached 19mm, where once again the image was out of focus.
From looking at the images, we believe the best focus distance is around 20mm.
At 21.0mm the image is clearly out of focus
21.0mm_focus_image.png:
The projection of this image then has wide lines and a larger standard deviation as the plot below shows:
21.0mm_focus_projection.png:
We wrote a code on python to analyse all of the frames. We first chose a random frame to project onto the x axis; this gave us the spectrum. By looking at the image, we then selected a slice from the data just including one peak. We then fit a Gaussian to the data; this gave us the value of the stdev and the position of the peak (the mean). We programmed a loop which found the stdev of all the frames, and plotted these stdevs as a function of the focus distance d in mm. This plot can be seen below.
Focus_sigma.png:
As you change the focus, not only will the width of the peak (stdev) change, but also the amplitude of the peak. When in focus, the image will be brighter and will therefore correspond to a higher number of photoelectrons (N_pe) and therefore a larger amplitude. As the image begins to defocus, the lines will be dimmer, corresponding to less N_pe and therefore a lower amplitude. Therefore to increase the accuracy of this plot, we decided to plot stdev - amplitude on the y axis as a function of focus distance d, as they are both changing with d. This plot gave us a more accurate indication of the best focus distance. This plot can be seen below.
Focus_sigma_amp.png:
As you can see for this plot (where we examined the far peak on the right), the best focus distance was d = 19.9mm, this image can be seen below,
19.9mm_focus_image.png:
This gave a better projection with a much smaller standard deviation as shown below.
19.9mm_focus_image_projection.png:
However, what we found interesting was that by observing another peak (the far right peak, for example), the plot showed that the best focus distance was d=20.1mm?
Angle
We then repeated this same procedure but this time changing the camera angle; this changes the orientation of the 3 lines. Once again, both the stdev and the amplitude change with the camera angle.
The image below shows the image with a small camera angle of 2mm, this is clearly out of line.
2.0mm.png:
The projection of this then has a wide standard deviation as shown below.
2.0mm_projection.png:
We initially plotted standard deviation as a function of camera angle in mm on the micrometer.
Angle_sigma.png:
The standard deviation minus the amplitude was then plotted against the roatation angle of the camera similar to the focus plot above.
Angle_sigma_amp.png:
Below an image of the straighter lines at 4.6mm can be seen.
4.6mm.png:
The projection of this then has a smaller standard deviation as shown below.
4.6mm_projection.png:
2. Errors
Now we have both plots, we need to find the errors. For the focus distance and camera angle, the error on these is just the smallest division on the micrometer; this is 0.01mm.
As we are using curve_fit on python, we can find 1 stdev errors on the parameters that curve_fit guessed to fit the gaussian to our data. These parameters are the amplitude, the mean, the stdev and the background. By using pcov (the estimated covarience of popt, where popt are the optimal parameters that curve_fit guessed), we can find the errors (to 1 stdev) on these parameters by doing perr =np.sqrt(np.diag(pcov)). We then selected the errors in the amplitude and stdev separately by picking out these elements from the 1D array. We then attempted to propagate these errors and plot them.
3. Further Data Taking
After this, we then changed the angle of the grating in an attempt to find further spectral lines. Considering we started the day at a grating angle of 3mm, we took numerous frames between the range between 0mm and 10mm, to explore both directions in our search for more spectral lines.
We found no further spectral lines below 3mm, but we found a couple of faint lines and 1 bright line above 3mm and below 10mm.
A few bright lines can be seen in the image below at 4.0mm.
4.0mm.png:
However, we then changed the exposure time and the screen stretch (SS) values. The SS changes the shades of gray of the image, allowing us to see parts of the image that we would not normally be able to see, such as faint spectral lines.
Min value of SS: 126.25
Max value of SS: 272.32
exposure time t=0.8s
At an angle grating of 2mm, we found 2 faint spectral lines we did not see before. We therefore took 5 frames at 2mm and t=0.8s.
At 4mm, we found another 2 fainter spectral lines. Again, we took 5 frames at this configuration.
At 5mm, we found another faint line and numerous very faint lines. We therefore increased the time exposure to t=1.2s and took 5 frames, in the hope that when these images are superimposed, the background is reduced and the signal will be more significant.
At 6mm, we found another relatively bright line - because of this, we only took 1 frame.
At 8mm, we found at least 3 very faint lines in the frame. We took 5 frames at 8mm and t=1.2s.