To verify the changes made to the model, let's compare it to the old model.

Impinged Power

The new model calculates the $P_\mathrm{KID}$ of the $j$th channel as a Riemann sum approximation of

$$\begin{equation} P_{\mathrm{KID},j} = \int_0^\infty\mathrm{PSD_{filter}}\left(\nu\right)\eta\left(j,\nu\right)d\nu \end{equation}$$

whereas the old model approximated this integral as a square filter with width $\Delta\nu=\mathrm{FWHM}$. Therefore the new model takes in a much wider range of the $\mathrm{PSD}$ and is thus expected to be higher in local minima and lower in local maxima of the $\mathrm{PSD}$.

The old and new power per channel, overlayed over the Power spectral density. The left y-axis shows power while the right y-axis shows the PSD. The simulated filters are hidden by default.

The figure above behaves exactly as expected.

Noise Equivalent Power

The $\mathrm{NEP}$ is calculated using the Riemann sum of the following integral

$$\begin{equation} \mathrm{NEP}_{\tau=0.5\mathrm{s},j}=2\int_0^\infty h\nu\eta\left(\nu,j\right)\mathrm{PSD} + \eta\left(\nu,j\right)^2\mathrm{PSD}^2 + \frac{2\Delta_\mathrm{Al}}{\eta_\mathrm{pb}}\eta\left(\nu,j\right)\mathrm{PSD}d\nu \end{equation}$$

where the three separate terms are the Poisson term, the bunching term and the recombination term respectively. Plotting this integral yields the following result:

The old and new Noise Equivalent Power per channel, overlayed over the Power spectral density. The left y-axis shows the NEP while the right y-axis shows the PSD. The simulated filters are hidden by default.

As we can see, and as is expected, the new model has a lower $\mathrm{NEP}$. Plotting the three separate terms, we get the following:

The three terms of the NEP plotted. Toggle in the legend to compare different terms.

As is clear from the figure, the terms behave as expected, with the new model 'smoothing' over the local extrema in the Poisson and recombination term, but leaving them otherwise unchanged. The bunching term is however smaller in the new model.

The Bunching Term

As we have seen the old approximation overestimated the bunching term by a factor of $\pi/2$. For a constant $\mathrm{PSD}$ The bunching term is given by

$$\begin{equation} \mathrm{NEP}_{\tau=0.5\mathrm{s},j,\mathrm{bunching}}=\frac{8}{\pi}\mathrm{PSD}^2\eta_0^2\mathrm{FWHM} \end{equation}$$

Whereas the old model this was given by

$$\begin{equation} \mathrm{NEP}_{\tau=0.5\mathrm{s},j,\mathrm{bunching}}=4\mathrm{PSD}^2\eta_0^2\mathrm{FWHM} \end{equation}$$

We can verify this in the following figure, where I have divided the old model by the new model. From our calculations, this should be on average $\pi/2$

The ratio between the old and new bunching terms. A constant line of half π is plotted as indication.

Simulated filterbank

In order to compare the simulated filterbank to the new and old models, it is more advantageous to look at the $\mathrm{NEP_{inst}}$. Recall that the $\mathrm{NEP_{inst}}$ is the $\mathrm{NEP}$ divided by the total instrument efficiency: comparing an 'unnormalized' $\mathrm{NEP}$ is deceiving, since a lower noise equivalent power does not necessarily mean a higher signal to noise ratio.

The instrument NEP of the old, new and newly simulated filters. The left y-axis shows the instrument NEP while the right y-axis shows the PSD.

Since the ideal filter model is generated with parameters which are the target for DESHIMA, the simulated filters behave almost without exception worse than the perfect Lorentzian filters. The outliers have very low transmissions, meaning that in order to improve the sensitivity the transmission needs to increase.

Noise Equivalent Flux Density

As described in the previous chapter, another modification to the model is the calculation of the $\mathrm{NEFD}$, both for a spectral and a continuum source, with the latter being twice the former.

Let's take a look at $\mathrm{NEFD_{cont}}$ first.

The Continuum Case

The different Noise Equivalent Flux Density for the continuum case. The left y-axis shows power while the right y-axis shows the PSD.

Because of the limited sampling range of the old model, the value of $\eta_\mathrm{sw}$ is (very close to) zero for channels where the atmospheric transmission is very close to zero. Because the new model takes the wide filters into account, this effect is lowered somewhat, since these channels are still loaded through the fringes of the Lorentzian when the atmosphere is opaque in the center.

The different atmospheric transmissions over the full filter band.

The Spectral Case

For the spectral case this difference is resolved, as spectral sources by definition don't load the fringes of a filter channel. This means that the atmospheric transmission isn't smoothed at its extrema.

The different atmospheric transmissions over just the bandwidth of the filters.

This in turn means that the $\mathrm{NEFD_{spec}}$ also approaches infinity at places where the atmospheric transmission approaches zero.

The different Noise Equivalent Flux Density for the spectral case. The left y-axis shows power while the right y-axis shows the PSD.

Bibliography