2.1.

Overview

TiEMPO is an end-to-end model, containing models of the astronomical source, the atmosphere, the telescope, the cryogenic instrument optics, and an integrated superconducting spectrometer with MKIDs (see Fig. 2). The details of the TiEMPO model can be found in Refs. 21 and 22, and the source code is publicly available.¹⁹ In the following we provide an overview of the modules, in the order of signal propagation from the astronomical source to the detector.

Fig. 2

System diagram of TiEMPO, showing each component with its input and output. TiEMPO depends on external packages ARIS,^15,16 ATM,¹⁴ and GalSpec.²⁰ TiEMPO outputs calibrated sky brightness temperature $T_{\text{sky}}$ and detector output $P_{\mathrm{MKID}}$, which contains photon noise, quasiparticle recombination noise, and atmospheric noise. The output timestream data can be analyzed by post-processing software such as De:code.¹³

2.2.

Spectrum of a Dusty High-Redshift Galaxy

The galaxy spectrum was created using the GalSpec package,²⁰ which we distribute as an open-source Python package that can be used outside of TiEMPO. Our goal is to create a galaxy template that is similar to the types of galaxies that we will be trying to detect with DESHIMA and with which we are able to model the potential future science cases. As such, we use an empirical approach to the creation of a galaxy spectrum, combining the continuum shape and spectral line luminosities from recent studies of observed local and high-redshift galaxies. The continuum is based on the two-component modified-black body fit to 24 galaxies at $z>2$ with Herschel (250, 350, and $500\mu \mathrm{m}$) and SCUBA-2 ($850\mu \mathrm{m}$) fluxes.^23,24 Here, we normalize the spectrum to the total far-infrared luminosity by integrating the spectrum from 8 to $1000\mu \mathrm{m}$. The spectral lines are simulated with a more creative approach that can be tailored to specific science goals. Relatively shallow observations, aimed at detecting atomic lines and CO, are simulated using spectral line luminosity scaling relations. Here we use the scaling relations from Ref. 25 for atomic lines and Ref. 26 for both CO and [CI] lines. The scaling relations of Ref. 25 are based on local star-forming and ultra-luminous infrared galaxies and high-redshift submm galaxies, whereas the scaling relations of Ref. 26 are based mostly on local galaxies. Deeper observations might resolve more complex molecular lines in both emission and absorption, such as ${\mathrm{H}}_2\mathrm{O}$, HCN, ${\mathrm{HCO}}^{+}$, ${\mathrm{CH}}^{+}$, NH, ${\mathrm{NH}}_2$, ${\mathrm{OH}}^{+}$, and HF. These species are only incidentally seen at high redshift (e.g., Refs. 27 and 28), and thus we rely on the line detections in the nearby ULIRG Arp 220 to supplement our spectrum for these molecular species.²⁹ Here, instead of scaling to the far-infrared luminosity, we scale the line brightness to the observed continuum at the line’s frequency. For this complete galaxy spectrum, we note that the brightness of these spectral lines is a probe of the conditions of the interstellar medium. As such, the line brightnesses (and even the continuum) are known to vary by up to 1 dex from source to source, which must be taken into account when applying for the necessary observation time or detection limits. Throughout this study, we assume a uniform line velocity width of $600\mathrm{km}{\mathrm{s}}^{-1}$ (full width half maximum), which is found to be the mean for typical submillimeter galaxy³⁰ and in line with recent observations of South Pole Telescope and bright Herschel sources.^31–33

2.3.

Creation of the Atmosphere Screen

Our goal here is to obtain the line-of-sight transmittance of the atmosphere ${\eta }_{\mathrm{atm}}$, which depends on the time $t$ and telescope pointing angle ($\theta$, $\varphi$). Here, $\theta$ and $\varphi$ are the elevation and azimuth angles of the telescope pointing, respectively. We start from the observation that the mm–submm ${\eta }_{\mathrm{atm}}$ at zenith ($\theta =90\mathrm{deg}$) is correlated with chiefly one variable, the precipitable water vapor (PWV).³⁴ Water vapor not only absorbs the submm waves but also introduces an extra path length (EPL) that is dependent on the line-of-sight PWV.³⁵ ARIS^15,16 uses a set of spatial structure functions³⁶ to produce a phase screen, i.e., a two-dimensional map of EPL as shown in the left panel of Fig. 3. Input parameters of ARIS include the inner and outer scales and exponents, the spatial size and resolution of the screen, and the root-mean-square (RMS) fluctuation of EPL in combination with a distance. Further details about the parameters can be found in Ref. 16, and typical values for the Atacama Large Millimeter–Submillimeter Array (ALMA) site are reported in Ref. 17.

Fig. 3

(a) Colormap of the output of ARIS for a $32\mathrm{m}\times 32\mathrm{m}$ sky window converted to PWV with Eq. (1). (b) The truncated Gaussian that is used as the telescope beam shape in the model. The volume of the Gaussian is normalized to unity, and it is truncated at a radius of 5 m, where its height is 10% of its peak height (Video 1, Mp4, 1070 KB [URL: https://doi.org/10.1117/1.JATIS.8.2.028005.1]).

TiEMPO converts an EPL screen to a PWV screen, using the following relation derived from the Smith–Weintraub constants³⁷ of EPL and the ideal gas law (see Ref. 21 for details), which is given as

Here, $k_2=70.4\pm 2.2\mathrm{K}{\mathrm{mbar}}^{-1}$ and $k_3=(3.739\pm 0.012)·{10}^5{\mathrm{K}}^2{\mathrm{mbar}}^{-1}$ are the Smith–Weintraub constants,³⁷ $\rho =55.4·{10}^3\mathrm{mol}{\mathrm{m}}^{-3}$ is the number density of molecules in liquid water, $R=8.314·{10}^{-2}{\mathrm{mbar}\mathrm{m}}^3{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$ is the gas constant, and approximately $T=275\mathrm{K}$ is the physical temperature of the atmosphere. Note that $d\mathrm{EPL}$ and $d\mathrm{PWV}$ are differences from arbitrary mean values of the optical path length and PWV, respectively. In TiEMPO the user can specify a mean PWV, around which the PWV fluctuates according to the ARIS-modeled EPL and Eq. (1). The mean PWV can be set to a constant or a vector that represents a slowly changing weather condition. The created PWV screen moves spatially in one direction, at the user-specified wind velocity, assuming that the structure of phase fluctuations is invariant when the atmosphere moves with the wind.³⁸ See Fig. 3(a) and Video 1 for an example of the moving atmosphere created in TiEMPO.

2.4.

Sampling of the Atmosphere by the Telescope Near-Field Beam

The water vapor in the atmosphere above the Atacama Desert is contained mostly in the layer up to $\sim 1\mathrm{km}$ from ground,³⁹ which is well within the near field of the telescope. Therefore, the beam pattern at this height has a similar pattern to the power distribution over the primary mirror of the telescope. In this study, we assumed that the PWV screen is 1 km above the telescope. For simulating DESHIMA on Atacama Submillimeter Telescope Experiment (ASTE) using TiEMPO, we assumed a Gaussian power pattern as shown in Fig. 3(b), which drops to $-10\mathrm{dB}$ at the telescope radius of 5 m. The PWV map created in Sec. 2.3 is filtered with this beam pattern, so the received power from the atmosphere is a weighted average within the beam. The user may include an arbitrary beam pattern in TiEMPO when detailed information is available from the design or measurement.

2.5.

Far-Field Beam of the Telescope

The far-field telescope beam is modeled by two properties: the main beam solid angle ${\mathrm{\Omega }}_{\mathrm{MB}}$ and the main beam efficiency ${\eta }_{\mathrm{MB}}$.^1,40 ${\mathrm{\Omega }}_{\mathrm{MB}}$ represents the solid angle of the beam excluding the side lobes. ${\eta }_{\mathrm{MB}}$ is the fraction of the beam contained in the main beam, out of the total reception pattern. The (total) beam solid angle, with the side lobes included, is then given by

We use the beam solid angle to define the effective aperture area as

2.6.

Radiative Transfer

For calculating the single-mode radiation transfer from the astronomical source to the detector, a subset of the DESHIMA-sensitivity software⁴¹ was used. Each component in the optical chain is modeled with a black body power density and a transmission factor ${\eta }_i$. The single-mode power density of a black body is equivalent to the Johnson–Nyquist noise and is given by

Here, $h$ is the Planck constant, $f$ is the frequency, $k_{\mathrm{B}}$ is the Boltzmann constant, and $T$ is the physical temperature of the emitter.⁴² The spectral power before the detector is computed by cascading the radiation transfer of each component as

2.7.

Spectrometer

TiEMPO is able to model any direct-detection (imaging-)spectrometer that couples the wideband input power into one or more spectral channels. Examples include integrated filterbank spectrometers that use a filterbank^1–3 or an integrated grating^4,5 and optical grating spectrometers. Currently, TiEMPO takes five spatial pixels (a center pixel pointed toward the astronomical source, and top, bottom, right, and left pixels viewing only the atmosphere) to simulate position-switching observations, but the pixel count can be increased to model multi-pixel imaging arrays. If the number of spectral channels per pixel is set to unity, then the model can represent a monochromatic imaging camera.

TiEMPO can import the spectral response of each detector, obtained from the design⁴³ or a measurement.⁴⁴ Here, we assumed a simple Lorentzian-shaped spectral transmission, which is a good approximation for a filterbank channel,² or a detector behind an optical (or substrate-integrated) grating.⁵ The frequency dependence was implemented by dividing the frequency range of 210 to 450 GHz (10 GHz wider on each side than the nominal DESHIMA 2.0 band, to take into account power coupling from outside of the band) into 1500 bins. The resulting efficiency is used to compute the power density with the radiation transfer equation, Eq. (7). Finally, the power in each bin is calculated as

Note that the $P_{\text{bin}i}$ in Eq. (8) is the expected (i.e., noiseless) value of the power. To calculate the frequency-integrated power detected by each detector at each moment $t$, $P_{\mathrm{MKID}}(t)$, we must consider noise (see Sec. 2.8).

2.8.

Detected Power and Noise

The best possible sensitivity of a pair-breaking detector like an MKID is set by the photon noise and quasiparticle recombination noise. The commonly-used narrow-band approximation for the noise equivalent power (NEP) limited by photon- and recombination-noise is given by¹

Here, $\overline{P_{\mathrm{MKID}}}$ is the expected value of the power absorbed by the MKID, ${\mathrm{\Delta }}_{\mathrm{Al}}=188\mu \mathrm{eV}$ ($\sim 91\mathrm{GHz}·h/2$) is the superconducting gap energy of aluminum, and approximately ${\eta }_{\mathrm{pb}}=0.4$ is the pair-breaking efficiency.⁴⁵ In TiEMPO we use the more general, integral form of the NEP⁴⁶ to take into account a frequency-dependent optical efficiency over a wide bandwidth, for each detector of the spectrometer. This method also accounts for photon-bunching of the wideband signal. Because the fluctuations in energy in different spectral bins are uncorrelated,⁴⁷ we can calculate the standard deviation in power for each spectral bin per sampling rate ($1/f_{\text{sampling}}$) from

Note that this is equivalent to calculating the detector NEP directly from the integral, which is expressed as

The integration over a wide bandwidth taking the filter spectral transmission into account is especially relevant for spectral channels that are near strong emission lines and absorption bands of the atmosphere.¹⁰

2.9.

Sky Temperature Calibration

After computing the noise-added power that is measured by the MKIDs, we want to relate this back to the original signal from the sky. We do this by expressing the received power in sky brightness temperature $T_{\text{sky}}$: the physical temperature of a black body that would have the same intensity as the semitransparent sky.¹ To this end we take the Johnson–Nyquist equation, which is given by

To relate the MKID power $P_{\mathrm{MKID}}$ to $T_{\text{sky}}$, TiEMPO internally performs a skydip simulation¹⁰ using the DESHIMA-sensitivity⁴¹ script. A skydip is a series of measurements in which the telescope “dips” from a high elevation (pointing at zenith) to a low elevation (pointing almost horizontally). When the elevation is lower, the telescope looks through a thicker layer of atmosphere, increasing the opacity and hence the power and the sky temperature, allowing us to construct a relationship between the two. In our simulations, we use elevation values in the range of $\theta =20\mathrm{deg}$ to 90 deg. The power and sky temperature data are interpolated for each channel and saved in the model. TiEMPO reuses these interpolation curves, so they only need to be created once. For further details on the numerical skydip calibration, see Ref. 21.