Intensity Interferometry Observations of the Hα Envelope of γCas with MéO and a Portable Telescope

An intensity interferometer correlates the intensity fluctuations of starlight between separated telescopes in order to measure the squared visibility. For two telescopes with a projected baseline r between them, the second-order coherence function is

$$\begin{eqnarray}&&{g}^{(2)}(r,\tau )=\displaystyle \frac{\langle {I}_{1}(t){I}_{2}(r,t+\tau )\rangle }{\langle {I}_{1}\rangle \langle {I}_{2}\rangle },\end{eqnarray} \tag{ 1 }$$

where I₁ and I₂ are the intensities recorded at each of the two telescopes, τ is the relative time lag between the signals, and the brackets indicate an average over time t. The Siegert relation (Siegert 1943; Ferreira et al. 2020) relates the second-order coherence function g⁽²⁾ to the first-order coherence function g⁽¹⁾ by

$$\begin{eqnarray}&&{g}^{(2)}(r,\tau )=1+| {g}^{(1)}(r,\tau ){| }^{2},\end{eqnarray} \tag{ 2 }$$

where the first-order coherence function can be separated into spatial and temporal components

$$\begin{eqnarray}&&{g}^{(1)}(r,\tau )=V(r){g}^{(1)}(\tau ),\end{eqnarray} \tag{ 3 }$$

where V(r) is the interferometric visibility of the source, given by the Fourier transform of the source sky brightness distribution. For an unresolved point-like source V(r) = 1, and the resulting second-order coherence function will depend only on the temporal component g⁽¹⁾(τ) given by the Fourier transform of the measured-light spectral density (Wiener 1930; Khintchine 1934). For linearly-polarized thermal light at zero optical path delay g⁽¹⁾(τ = 0) = 1, where, for time lags much greater than the coherence or correlation time, the first-order coherence function should be equal to zero such that there is a "bunching peak" centered about zero optical path delay with an effective temporal width given by the coherence time. The peak amplitude at zero time lag of g⁽²⁾ thus measures the squared visibility at some projected baseline assuming that the instrumental resolving time is shorter than the light coherence time. The coherence time can be defined by the integral of the squared first-order coherence function (Mandel & Wolf 1995)

$$\begin{eqnarray}&&{T}_{c}=\int | {g}^{(1)}(\tau ){| }^{2}d\tau =\int | s(\nu ){| }^{2}d\nu ,\end{eqnarray} \tag{ 4 }$$

which is equal to the integral of the squared normalized spectral density s(ν) by Parseval's theorem. For visible light with a bandpass of Δλ ∼ 1 nm the corresponding coherence time is of order 1 ps, much shorter than what can be achieved with conventional detectors. In this case, a measurement averages over many coherence times and reduces the value of the g⁽²⁾ peak amplitude at τ = 0 by a factor of ∼T_c /T_d where T_d is the effective time resolution of the detector. The amplitude of the g⁽²⁾ peak therefore measures this loss of contrast times the squared visibility. The squared visibility can be extracted by dividing the value of g⁽²⁾(r) − 1 peak amplitude measured between telescopes to the g⁽²⁾(r = 0)−1 peak amplitude measured at the zero baseline under the assumption that the profile of the g⁽²⁾(τ) peak is constant. In practice, we measure the ratio of the area of the g⁽²⁾(r) − 1 peak to the area of the g⁽²⁾(r = 0)−1 peak for the squared visibility

$$\begin{eqnarray}&&| V(r){| }^{2}=\displaystyle \frac{\int \left({g}^{(2)}(r,\tau )-1\right)d\tau }{\int \left({g}^{(2)}(r=0,\tau )-1\right)d\tau }.\end{eqnarray} \tag{ 5 }$$

The denominator, or equivalently the area of the g⁽²⁾ peak at zero baseline, corresponds to the coherence time that can be calculated from the measured spectrum as given by Equation (4). The equivalence between the coherence time from intensity interferometry and spectral measurements assumes that the spectral resolution is narrower than any spectral lines within the instrumental bandpass. Since intensity interferometry measurements probe the intrinsic spectrum it is a useful method for characterizing narrow spectral lines present in, for example, studies of light scattering off of atomic clouds (Dussaux et al. 2016), with potential applications in astrophysics (Tan & Kurtsiefer 2017).

The II observations presented in this paper were performed by outfitting two telescopes located on the Calern Plateau site of the Observatoire de la Côte d'Azur with individual coupling assemblies (CAs). A CA is mounted near the focus of each telescope, both shown in Figure 1. The first facility was the 1.54 m diameter MéO (Métrologie Optique) telescope primarily used for satellite laser ranging (Bertrand et al. 2021), lunar ranging measurements (Bourgoin et al. 2021), and low-Earth orbit satellite laser communication (Giggenbach et al. 2022). The optical design is based upon a Ritchey–Chrétien configuration on an altitude-azimuth mount with a primary focal length of 31.0 m giving an approximate focal ratio of f/20.1. In typical operation the light is brought to a Coudé focus, but for II observations the light is redirected to the CA along the Nasmyth arm using a removable 45° mirror. In addition, a f = 150 mm lens is inserted before the CA in order to decrease the effective focal length.

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The second facility is the portable 1 m diameter T1M, a Newtonian telescope on a Dobson-type fully motorized azimuthal mount. The portability of the telescope enables configurable baselines to expand the accessible coverage of the uv plane where the telescope can be disassembled and moved in just a few hours. The telescope has a primary focal length of 3 m, and a Barlow lens is included in order to expand the effective length at the input of the CA.

While both telescopes are azimuthal, there will be a relative field rotation due to the Newtonian versus Nasmyth optical designs. However, each CA utilizes polarizing filters that must be aligned with respect to one another, although not with respect to the target as astrophysical polarization effects are not investigated. To compensate, the MéO telescope CA is mounted into a rotation stage that orients the CA such that the polarization axes are aligned. The stage is actively controlled throughout the observation where the amount of rotation is determined from the target sky position.

The relative position of both telescopes must be known with a precision less than a few centimeters for optical path delay corrections. For MéO, the position was previously determined to millimeter accuracy in terrestrial coordinates due to its use in geodetic surveys. To determine an absolute position of the mobile T1M telescope, geodetic markers were installed by the National Institute of Geographic and Forest Information at the ground level for several positions and their positions were measured from differential GPS methods. The T1M was installed above these markers and the offset between the marker and the T1M reference point was estimated. The estimated cumulative error on the reference position is ±1.5 cm in all directions.

The primary function of the CAs (shown in Figure 2) is to perform spectral/polarization filtering, and fiber injection. The current version includes an automated tip-tilt device that provides stable fiber injection over several hours without manual intervention (Matthews et al. 2022). The detector signal output is fed to a time-to-digital converter (TDC) that measures photon arrival times, and produces the correlation between all relevant pairs of detectors. Across both telescopes, there are four independent measurements of the zero baseline correlation g⁽²⁾(r = 0, τ) enabled by using fiber splitters in each polarization mode. These allow normalization of the spatial correlations across telescopes to measure visibilities. Spatial intensity correlations are obtained by calculating the correlation across telescopes for all pairs of detectors in the same polarization mode corresponding to a total of 8 measurements of g⁽²⁾(r, τ). During one night, a time delay monitoring system was used on the electronic cables connecting the detectors from the MéO telescope to the TDC. The drift throughout the whole night was significantly less than the characteristic jitter of the detectors of ∼500 ps.

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The coherence time obtained from zero baseline intensity correlations was compared to expected values from the spectral throughput. These temporal intensity correlation functions were computed for each polarization state and for each telescope by summing all individually acquired correlations acquired over the entire observation. Each of the resulting correlation functions were then shifted by the instrumental delay and then coadded together. The resulting correlation function is displayed on the left in Figure 3. The peak is fit by a Gaussian with free parameters for the amplitude and width. The coherence time, given by the integral of the peak, is extracted via the fit values. Before fitting, there is a choice of the time-lag range to fit over and the number of time lags to bin. Here, we fit and show the data over a range of ±20 ns binned into 50 ps bins. Under these parameters, we find an amplitude of (1.43 ± 0.05) × 10⁻³ and a full width at half maximum of 885 ± 40 ps corresponding to a measured coherence time of 1.35 ± 0.05 ps. Systematics of the fitting process were studied by fitting the data varying the fit range from ±10 to ±40 ns and additionally the binning size from 10 to 80 ps. Within these parameters, we find a maximum difference of 0.015 ps in the extracted coherence time—notably less than the measurement error in the previously quoted coherence time; the correlations from different polarization states and telescopes were coadded and subsequently fit. This procedure requires that shape of the bunching peak in each correlation, given by the temporal response of the detectors, are similar. To test systematics, each correlation function for both polarization states and telescopes were fit by a Gaussian to extract the coherence time. Each individual fit was within 1σ of the quoted coherence time, and furthermore the weighted mean of fits (1.35 ± 0.05 ps) is in perfect agreement with a single fit of coadded correlations indicating that within our measurement precision there are no significant systematics that preclude us from combining individual zero-baseline correlations from separate detector pairs.

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The coherence time measured from the zero-baseline correlations can then be compared to expectation from the recorded spectrum. The spectral transmission of the Hα filter was measured in the laboratory with a high-resolution spectrograph (Matthews et al. 2022). Spectra of γ-Cas with resolution R = 25,000 were recorded contemporaneously with our observations using a Whoppshel echelle spectrograph, provided through collaboration with the 2SPOT ⁷ association of amateur astronomers. This is especially important as the width of the temporally variable emission line is narrower than the filter bandpass, thus affecting the coherence time. The right side of Figure 3 shows the filter transmission, and the emission line spectra. Through Equation (4) we extract the expected coherence time to be 1.41 ps, which is 1.2σ larger than the value from zero-baseline correlations when taking only the measurement uncertainty and thus in fair agreement. In contrast, the coherence time that would be expected for a flat stellar spectrum would be 1.16 ps. This is considerably less than the measured value by 3.8σ illustrating the importance of including the emission line profile in calculations of the coherence time. The general agreement of the coherence time measured between intensity interferometry and spectral measurements indicate that there are no systematic effects arising from the presence of unidentified narrow spectral lines due to a lack of spectral resolution.

The spatial intensity correlations correspond to the correlations between all detector pairs on separate telescopes that are observing in the same polarization mode corresponding to a total of eight cross-correlations across telescopes. All computed correlations are shifted in time by instrumental and geometrical delays, and then coadded together. The full data set corresponds to a wide projected baseline and position angle range and so the data was divided into smaller baseline ranges. For each subdivision, we compute the averaged correlation function and then fit a Gaussian function to the resulting bunching peak. The measured areas of the g⁽²⁾ peak for each subdivision are presented in Table 1. Squared visibilities are extracted by computing the ratio of the integral of the Gaussian peak of the cross-correlation to the computed value from the measured spectrum.