# 4. Efficiency¶

It is important to calculate the efficiency of the final spectral extraction from the SEDM pipeline. This is done by comparing the expected standard star flux with the observed standard star flux. Although that sounds easy, in practice there are many pitfalls to this calculation. This page documents the process by which we derive the efficiency.

At this time it is important to acknowledge the efforts of Jason Fucik, Rich Dekany, and David Hover in producing the models of the SEDM optical system and in performing lab measurements and analysis. In particular, Jason made the crucial discovery of the low throughput ‘smoking gun’ and is leading the acquisition of an improved micro-lens array.

## 4.1. Standard Flux¶

The reference data used for this calculation is a tabulated list of fluxes for a given standard star. In our case these are in units of $$F_{\lambda,ref} = erg\ s^{-1} cm^{-2} A^{-1}$$. Since CCDs measure the number of photons observed, our first task is to convert this energy flux into photon ($$e^-$$) flux, $$F_{ph,ref} = e^-\ s^{-1} cm^{-2} A^{-1}$$.

We know the energy of a photon is $$E_{ph}(\lambda) = hc/\lambda\ erg/ e^-$$. The wavelength dependency must be accounted for when converting to photon flux. Planck’s constant is $$h = 6.62606885\times 10^{-27} erg\ s$$ which, when multiplied by the speed of light in $$A\ s^{-1}$$ gives $$hc = 1.98782\times 10^{-8} erg\ A$$. Thus, we get a conversion to photon flux of $$F_{ph,ref} = \frac{F_{\lambda,ref}}{E_{ph}(\lambda)} = \frac{erg\ s^{-1} cm^{-2} A^{-1}\ e^-}{1.98782\times 10^{-8} erg\ A}\lambda\ A$$. This gives a final conversion of $$F_{ph,ref} = 5.034\times 10^7 F_{\lambda,ref} \lambda$$, in units of $$e^-\ s^{-1} cm^{-2} A^{-1}$$.

## 4.2. Observed Flux¶

While the previous calculation is pretty basic, calculating the observed spectrum is where many pitfalls are encountered. In particular, since our standard flux is now in $$e^-\ s^{-1} cm^{-2} A^{-1}$$ and it is very unlikely that the native wavelength bins of our observations are all 1 Angstrom wide, we must carefully account for the native size of the wavelength sampling. To do this, we must use the wavelength scale that is most representative of the cube’s geometry solution.

It is also important that we collect as much of the standard star’s light from the raw image as possible. This means using a large number of spaxels, which makes our calculation more susceptible to background problems. Because of this, we derive our fiducial wavelength scale from the central five arcseconds of the IFU field of view. This avoids any edge effects and should represent the region where most of the standard stars are observed.

In addition, all observations of standard stars are corrected for atmospheric extinction using the standard extinction curves for Palomar (Hayes & Latham 1975).

### 4.2.1. Wavelength Scale¶

Previously we used a function $$(\lambda(x) = 1050(239/240)^x, x: 0 - 265)$$ for the fiducial wavelength scale that roughly approximates the run of wavelengths for SEDM. However, it is important for this calculation that the wavelength scale be as close to the native scale as possible. Therefore, we switched to using an average scale derived from spaxels in the central five arcseconds of the IFU. This changed the shape of the efficiency curve and made it more accurate.

### 4.2.2. Wavelength Bins¶

Next we must calculate the size of the wavelength bins, given as $$\Delta\lambda(\lambda)$$. Most spectrographs have a native wavelength bin size that is approximately constant, which is another way of saying the dispersion is fairly constant. Since the SEDM was designed not to have a constant dispersion, but to have a constant resolution of $$R = \lambda/\Delta\lambda \approx 100$$, the dispersion varies substantially over the wavelength range. Thus, our wavelength bins are a function of wavelength, as shown below.

The difference between the average and the function-generated wavelength scales becomes more pronounced when you use them to calculate the wavelength bins. This was the main reason we chose to use an average wavelength scale rather than a functional form. Thus, our observed flux is $$F_{ph,obs} = e^-\ s^{-1}\ \Delta\lambda(\lambda)^{-1}$$, where $$\Delta\lambda(\lambda)$$ is given by the average wavelength bins.

## 4.3. Effective Area¶

Now we can calculate the effective area of our instrument using the following formula: $$A_{inst,eff}(\lambda) = F_{ph,obs} / (F_{ph,ref} \Delta\lambda(\lambda))$$. All units cancel (see previous formulae) except $$cm^2$$, giving what is called the effective area of the instrument as a function of wavelength.

## 4.4. Efficiency Ratio¶

We can now compare this area to the actual area of the telescope, corrected for reflective losses to derive the efficiency, using this formula: $$T_{inst}(\lambda) = A_{inst,eff}(\lambda)/A_{tel,eff}$$.

For reflective losses of the P60 telescope, we have two mirrors, the primary and secondary, that contribute. If we assume an average of 91% reflectivity for the primary (from reflectometer measurements after the September 2017 re-aluminization) and somewhat less for the secondary which has not yet been re-aluminized, say 90%, we get the total reflective throughput of 82%. We then use the area of the primary mirror, accounting for the secondary obstruction, or $$A_{tel} = 18,000\ cm^2$$. Multiplying the reflective througput times the telescope area gives a telescope effective area of $$A_{tel,eff} = 14,742\ cm^2$$. This is what we divide our instrumental effective area curve by to get the instrumental throughput.

Below is the figure that shows a typical efficiency curve using our best understanding of the calculation and of the instrumental wavelength bins. The efficiency is only as good as the quality of the night and represents an approximation of the true efficiency. Clouds will tend to lower the measured efficiency, but moonlight or other sources of anomolous background will tend to raise the measured efficiency. In general, clouds are a more common source of efficiency offset and so most measurements are lower limits on the true efficiency.

## 4.5. Importance of Wavelength Bins¶

To visualize how the efficiency is calculated, we can think of the instrument as having buckets to collect photons that correspond to the CCD pixels. The dispersive elements of the instrument define what the wavelength ‘size’ of each bucket is, while the spatial ‘size’ of each bucket is determined by the optical properties of the entire system (telescope + instrument). The efficiency is simply the number of photons we actually collected in our bucket divided by the number of photons we expect a given bucket to collect. The number of photons we expect to collect is obviously a strong function of the size of the bucket; the bigger the bucket the more photons we expect to collect. If we assume our buckets are smaller than they really are, our expectations are correspondingly low and we get a higher efficiency. The plot below shows the result of our initial efficiency calculation, assuming that all of our buckets are 1 nm in wavelength.

The shape of this curve shows a strong blue linear trend, which is contrary to the expected curve based on the optical design, ray-traced in ZMAX (see gray curve in figure six below). Once we discovered that we were not accounting for the wavelength bin size, we re-calculated the efficiency and since our wavelength bins were not 1 nm, but instead ranged from approximately 1.5 to 4.5 nm (green curve in figure two), the overall efficiency dropped considerably. The shape seems to be closer to what is expected based on ray-tracing. However, here we are still using the functional form for our fiducial wavelengths. The plot below shows how the overall efficiency dropped significantly, but because the wavelength bins now vary in size much closer to the native bin sizes, the shape changed to a more expected form.

If we examine the native wavelength solution we find that, in fact, the wavelength bins range from 1.5 to 5.7 nm (blue curve in figure two) and have a trend that differs from that generated using the functional form. Compare the figure above with the first efficiency curve and you will see that at 500 nm, the efficiency increases by 0.5%, but the peak efficiency goes down. If you compare the green and blue curves in figure two above, you can see that the largest differences occur around pixel zero and pixel 200. Referring to figure one above, it is apparent that this changes the shape of the curve primarily at the red and blue ‘shoulders’.

## 4.6. Ray Tracing and Lab Measurements¶

While the SEDM was in the lab, from March through August 2017, we were able to do some analysis of the instrument using a monochrometer and to analyze a ZMAX model of the optics. Below is a figure showing some of the results that we can compare with our on-sky measurements.

The yellow curve was derived the same way that figure four was calculated. Our best calculation shown in figure three has a shape closer to the gray curve, but with a lower peak throughput by a factor of more than six. We point out that the gray curve is calculated for a single spaxel ray and does not account for losses due to the lenslet filling factor or dead zones between lenses. It is puzzling that our initial (and incorrect) calculation agrees so well with the lab throughput measurements shown by the red and blue curves. It is possible that there is still some accounting for wavelength bins in the lab measurements that needs to be done.

## 4.7. The Effect of Filling Factor on Efficiency¶

In our efforts to understand the low throughput of the current SEDM, we have tried to estimate the filling factor of the multi-lens array (MLA). In the manufacturing process for any MLA, a certain fraction of the array becomes unusable because of dead zones at the borders of the lenslets. In our analysis, we have found that it is crucial to keep these dead zones as small as possible because, not only do they represent a loss of light, but they are also a source of scattered light. The specification for the orignal MLA was to have a filling factor of around 95%. Our investigations have revealed that, due to a misalignment of the front and back lenses in the MLA, the effective filling factor is actually closer to 80%.

The impact of this low filling factor is rather extreme and may completely explain the low instrumental throughput. The filling factor enters the throughput calculation as a factor to the power of the number of spaxels involved, $$T_{inst} = T_{spax} \times f_{fill}^{N_{spax}}$$. Using the peak throughput predicted for a single spaxel (the grey curve in the lab measures figure) of 45%, a filling factor of 80%, and assuming we cover seven spaxels, we get $$T_{inst} = 0.45 \times 0.80^{7} = 0.09$$, which is very close to the measured throughput peak in the figure above of our best efficiency calculation. The remaining difference is likely due to the fact that our standard stars usually cover more than seven spaxels and thus the impact of the filling factor would be greater.

The impact of filling factor is also illustrated by the figure below.

### 4.7.1. Good News¶

Ultimately, this is good news for the prospect of improving the SEDM throughput. In particular, we can avoid the whole issue of alignment by making the MLA plano-convex instead of double convex. This should result in a filling factor much closer to 100% and thus our throughput should jump by nearly a factor of seven. We have indeed redesigned the MLA to be plano-convex and are working with a vendor with the goal of producing a MLA with a filling factor of 98% (light blue curve in the figure above) as a replacement MLA for SEDM (and for any future versions of the SEDM).

## 4.8. Efficiency Trend¶

As stated above, the quality of the night most typically reduces the efficiency measurement due to atmospheric extinction (clouds), but can also increase the efficiency if there is a high background (moon). The best way to mitigate these effects is to look at the trend over time. Below is a figure that shows the efficiency in wavlength bins over the course of the last 700 days. This was calculated after re-processing all the archival data with the average fiducial wavelength scale.

Several features of this plot stand out. There are short periods of higher efficiency that go against the general trend. These are most likely from observations of standard stars that have a high background due to moonlight.

The other feature is the increase in efficiency to a peak near JD 2457640 and then a general decline. It’s hard to understand the rise in this trend, while the decline is expected as the mirror coatings deteriorate. There was a lot of experimentation with the instrument configuration during the early days, although this would better explain jumps in efficiency and not a slow general trend.

We also see that the current efficiency is slightly lower than the peak from the previous group. If somehow the trend that was seen in the previous 300 days hold for the next 300, then we should increase to a similar if not higher peak.

Last updated on 08 March 2018