2.1

Field of view

With spider, the observed object is apodized by the “antenna lobe” in intensity L of the waveguide inside which the light is coupled, so that the effective object whose complex visibilities are measured by the interferometer is:

L is the intensity of the back-propagation towards the object of the waveguide’s propagation mode in the aperture plane: L = |FT(P)|². L is typically a Gaussian 2D function of FWHM λ/D, where λ is the wavelength and D is the diameter of the apertures, as studied in details in Refs. 13,14. Finally, the field-of-view (FOV) of spider is:

2.2

Visibility measurement model

Let us consider a pair of apertures separated by B measuring the spatial frequency (u, v) = B/λ. Let us call C₁ (respectively C₂) the coupling coefficient of aperture 1 (respectively 2) towards the output, ϕ_k the modulation phase of the k-th measurement, and N_ph the mean number of photons received in each of the two apertures, during a given exposure time. Assuming balanced channels i.e.,  , a model of the recorded raw data (intensities) is:

where the complex visibility of the observed object is:

The mean number of photons received and detected per aperture and exposure time is:

By using typically four measurements (ϕ_k = (k – 1)π/2, for k ∈ {1, 2, 3, 4}) followed by a demodulation^†, one obtains a complex measurement y(u, v) proportional to the mean number of photons received in each of the two apertures and to the complex visibility of the object at the measured spatial frequency:

Thanks to the Van Cittert-Zernike theorem:

so γ(0, 0) = 1. Equation 6 then yields y(0, 0) = N_ph, i.e.,  the collection y of complex measurements y(u, v) is a set of samples of the Fourier transform of o_eff(α_x, α_y) of integral (i.e.,  of total flux) N_ph.

2.3

Resolution and number of resolution elements

As for any interferometer, the maximum spatial frequency recorded by a compact interferometric imager is given by the maximum distance separating two apertures, or maximum baseline B_max, and is B_max/λ. The angular resolution of such an instrument is thus:

The angular field being λ/D, the number of resolved elements of a compact interferometric imager in the observable angular field is B_max/D.

2.4

Spectral width

In case of point-like apertures separated by B interfering at a wavelength λ with zero bandwidth, the visibility is measured at the spatial frequency B/λ. As soon as the bandwidth is non-zero, the detected interferogram is an average of inteferograms at the spatial frequencies given by all the wavelengths in the band. Therefore, what spectral bandwidth can be tolerated?

To answer this question, it is useful to realize that the recombined beams contain interferences between apertures of diameter D spaced by B and thus between pairs of points separated by distances ranging from (B−D) to (B+D). Such an interferometer with waveguide injection intrinsically measures an average of spatial frequencies of the object over a typical width of D/λ. It is then reasonable to specify that the spatial frequency averaging due to the spectral width dλ is less than the intrinsic averaging due to the non-zero aperture size. Differentiating f = B/λ we obtain the following condition:^‡

For the shortest wavelength and the longest baseline, this yields:

The inequality 9 can also be read as follows: the spectral resolution λ/dλ must be greater than the number of (spatially) resolved points of the interferometer, B_max/D.

3.1

Noise modeling for a compact interferometric imager

Under the assumption that |γ(u, v)| ≪ 1, the variance of the noise on the complex measurement y(u, v) of Equation 6, resulting from the demodulation of four raw measurements given by Equation 3, is^§:

where N_ph is still the average number of photons received in each of the two apertures contributing to the interference during a raw measurement. This directly follows from the fact that y(u, v) is the demodulation of 4 raw data measurements, where each is corrupted by photon noise, of variance N_ph.

The variance of these complex measurements, normalized by the square modulus of the signal at frequency (u, v), is therefore simply, according to Equation 5:

or:

3.2

Noise modeling for a conventional imager

The total number of elementary apertures of a compact interferometer spider is denoted N_d. Considering each elementary aperture used a single time and connected by pairs, N_d is necessarily even and the number of measured spatial frequencies is N_d/2. We consider an imaging system with the same collecting surface as this spider^⁋. The object observed by the imaging system is assumed to be the object seen by the SPIDER device, o_eff, to simplify the comparison. The measurement of each frequency requires the detection of 4 signals of average amplitude N_ph. The measurement of all these spatial frequencies therefore takes a total of N_phtot = N_d/2×4N_ph = 2N_dN_ph photons. Hence, the image model of a conventional imager collecting a total of N_phtot photons writes:

where h is the discrete PSF, and * is the discrete convolution operator.

We have already assumed that the visibility is negligible with regards to 1. For a classical imaging instrument, we additionally assume that the scene luminance is sufficiently homogeneous, so that the noise can be assumed to be stationary, i.e.,  so that the variance of the noise of the classical imager can be assumed to be approximately constant over all N_pix pixels. Hence, in the Fourier space, the noise spectrum is uniform and equals N_phtot.

Since , as pointed out in paragraph 2.2, in the Fourier domain, the spectrum of the noise normalized by the square modulus of the signal at frequency (u, v) is:

or:

Comparison of Equations 11 and 13 suggests that signal-to-noise ratios are similar for both types of instruments, i.e., proportional to , and that they are identical if:

at all frequencies except the zero frequency, at which the MTF is 1 by convention. In other words, with respect to noise propagation, a “standard” compact interferometric imager (standard in the sense that each spatial frequency would be measured once and only once) is equivalent to a conventional imager with a flat MTF of the order of . Note that focal-plane imagers with quite flat MTFs exist, namely phased array instruments or sparse aperture telescopes with minimal redundancy such as the ones discussed in Ref. 17. For these, the MTF is flat and of the order 1/N_tel, where N_tel is the number of apertures.

Finally, the above results show that, for the highest spatial frequencies, a compact interferometric imager and a conventional imaging instrument have a very similar behavior with respect to noise. For the former, the propagation of noise is unfavorable for the lower frequencies. However it is quite possible, for a particular task (detection for example), to design the transfer function of a compact interferometric imager to make it higher at the spatial frequencies relevant to the task.

4.1

Optimization of the frequency coverage of dense 1D arrays

The total number of apertures in a single arm (1D) is denoted N_p. Still considering each aperture used a single time and connected by pairs, N_p is even. The number of measured spatial frequencies is consequently N_p/2 and the measured frequencies have values in the range [1, N_p − 1].

A frequency coverage is defined as buildable if there is an aperture configuration that allows it to be measured (not all frequency coverages are buildable). Finding compact buildable frequency coverages composed of consecutive spatial frequencies {1, 2, …, N_p/2} is a problem formulated and solved in the framework of combinatorial theory by Skolem.²³ Such a frequency coverage is called a Skolem set and its associated aperture configuration a Skolem sequence.

More generally, it is possible to build frequency coverages in the form {n_min, n_min + 1, …, n_min + N_p/2 − 1}, where n_min is the first measured spatial frequency. This problem has been introduced by Langford.²⁸ Such a frequency coverage is called a Langford set and its associated aperture configuration a Langford sequence. The existence of buildable compact frequency coverages was proved for n_min = 1 by Skolem in 1958, for n_min = 2 by Davies²⁴ in 1959, then partially generalized by Bermond²⁵ in 1978 and finally completed by Simpson²⁶ in 1983.

4.2

Application to a SPIDER-like interferometer

Applying the Skolem results, an illustrative aperture configuration is provided in Fig. 4.

Figure 4:

top panel: example of a Skolem aperture configuration with 32 apertures - bottom panel: associated normalized frequency coverage ranging from 1 up to 16.

4.3

Suggestion of a hybrid architecture using Langford configurations

It is possible to further optimize the PRL obtained for the SPIDER-like design by considering an interferometer based on a Langford sequence, complemented with a small monolithic telescope measuring short spatial frequencies.

If buildable, the Skolem configuration reaches a maximum spatial frequency of N_p/2 by definition whereas the Langford configuration reaches a maximum spatial frequency of (3N_p − 2)/4. Because the maximum spatial frequency measured increases from N_p/2 to (3N_p − 2)/4, the resolution of this hybrid interferometer is therefore improved by about 50% with respect to the interferometer with the Skolem configuration.

Maintaining the SPIDER-like interferometer aperture complexity, i.e. N_p = 30, an illustration of this Langford aperture configuration is presented in Fig. 5. The represented aperture configuration produces a compact frequency coverage ranging from spatial frequencies 8 to 22, improving the frequency coverage provided by a Skolem configuration, i.e. by considerably stretching the highest spatial frequency.

Figure 5:

top panel: example of a Langford configuration with 30 apertures - bottom panel: associated frequency coverage ranging from spatial frequency 8 up to 22.

Finally, in Fig. 6b, we have represented the two-dimensional optical transfer function following the merging of the frequency coverages from the monolithic telescope and the SPIDER-like interferometer.

Figure 6:

optical transfer functions - a) SPIDER-like instrument with radially disposed arms, discrete spatial frequencies measured by the interferometer {1, 2, 3, 4, 5, 6, 8, 10, 13, 17, 22, 29} (black) - b) suggested hybrid instrument: monolithic telescope complemented by SPIDER-like instrument with radially disposed arms, continuous spatial frequencies measured by the telescope [1, 8] (red) and discrete spatial frequencies measured by the interferometer {8, …, 22} (black).