Determining instrumental parameters from a shuffle image
 

Here is a worked example of how we determine the TTF plate spacing, the order of interference, the free spectral range, the spectral bandpass, resolving power and effective finesse from a single shuffle image. The entire process takes only 5 minutes. These shuffled arc images are fundamental to monitoring TTF stability throughout the course of a night, every few hours or so.

First, switch in the calibration lamp mirror (chimney flap) and turn on your preferred calibration  source. I have found CuAr good for all the red filters, except Ne is preferred for R0. The expected lines are given here.

Now type the following. The colours refer to the TAURUS control  window or the CCD control window. Recall that the obeyw command is typed after hitting "." on the keypad. Of course, you can perform all the same actions with cursor clicks in the SMS control  window.

  

obeyw taurus focal ?           Select the  spectral region with an order sorting filter (e.g. B, V, R0)

obeyw taurus etalon ?         Ensure you have the correct TTF in the beam

obeyw taurus pupil  8           Ensure you have the clear aperture in the pupil wheel

obeyw taurus aperture  3    Ensure you have the slit in the aperture wheel

win mitll_shuffle80             Set the CCD window size in the CCD window (read speed irrelevant)
 
obj ?                                        Set the new object name to "CuAr arc; filter ?" or whatever

obeyw taurus run run_ccd run_step  0  7  553       The numbers are the etalon "z" values
 
 


How did we choose the run_step "z" values?  The TTF response is cyclical over a free spectral range "FSR" in the  same units. The "FSR" at H-alpha is 340;  the "FSR" at any other wavelength is simply proportional to wavelength. Without foreknowledge, simply scan the TTF through roughly two FSRs. In this way, we ensure there will be at least one set of calibration lines in order of increasing "z" in the spectrum. In the example given here, we do not go far enough to ensure this. For the sampling step, use FSR/40.

In the example above, the run sequence generates a single image, and here it is:

Figure: CuAr lamp through R1 (710/26) filter and red TTF at high resolution.

Within iraf, bin up the columns of the arc image f3 to improve signal to noise. Grab the central column and fit to it within an xgterm window.

This is what you should see:
 
 
Now we fit the profiles.  Don't be perturbed by the up-down chatter as f3 is made from charge shuffling an image of a slit. This is how we convert a tunable imager into a long-slit spectrograph - a highly inefficient one at that.

Place the cursor on the flat part on either side of the 4-line group and type "d". Mark each peak with "m", followed by "q", "a", "a", "n".  The AAO implementation of iraf (v. 2.11) includes the new splot which allows lorentzian fitting with "l" in place of "m". The fitted line centroids in units of "z" are given at the bottom. Cycle through them forwards using "+" or backwards using "-".  To exit, keep hitting "q".

Store the centroids in a file, say fit3:

and carry out the least-squares fit: (Alternatively, note that splot writes the fit parameters out to splot.log. Simply use this file after inserting the correct wavelengths in the first column and stripping any preamble.)  Notice that we have "wrapped" the faint line at y = 365.3 through a free spectral range. The first strong line falls at y = 275.7 and repeats at y = 589.7 so that FSR_y = 314. Another useful quantity used below is the fitted line width,  dy = 11.8. The fit yields:

          -7853.972      1.194301                                                          y_offset     dy/d(lambda)
          221.5427     0.0313787                                                           +/-error    +/-error

So now we have the lambda-y relation,  y =  1.1943 * lambda - 7854.0, to compute the fundamental constants of TTF. For lambda = 7100 A, 

 
order of interference m lambda / { FSR_y * d(lambda)/dy } 27.00 
plate spacing l_o  m * lambda / 2  9.6 um
free spectral range FSR_lam lambda / m  263 A 
spectral bandpass dlam dy * FSR_lam / FSR_y 9.8 A 
resolving power R lambda /  dlam 770
effective finesse N_eff  FSR_lam / dlam 27
 

Notice several things. The order of interference is a whole number!  The filter bandpass is a perfect match with FSR_lam. The plates are operating at a spacing of one fifth of a human hair thickness and they are 70mm in diameter!  The effective finesse looks a little low, and in fact it should be more like 40. However, the line profiles above are not Nyquist sampled, i.e. steps of FSR/80. 

Moreover, you should note that the instrumental parameters are entirely independent of the units of the y-axis. We could have used a sausage cube with either CCD frame numbers or etalon "z" values along the third axis with the same outcome.

For the swot, there are stacks of additional details one can add here. For example, at very small spacings, all of the above formulae need to be modified for reflectance phase (e.g. Jones & Hawthorn 1998, PASP). A full mathematical discourse gets into some beautiful mathematics which is the very underpinning of the burgeoning thin film industry. 

Here, we provide a tabular summary of what the arc lines look like for different filters and at different resolutions.

Time for coffee...