In an ideal nulling interferometer, the electric fields of the light from the collecting telescopes are combined with a prescribed set of amplitudes and phases that produce a perfect null response at the star (Figure 1). In this case the integration times needed for planet detection and spectroscopy depend on the planet signal strength and the level of photon shot noise (from local and exozodiacal emission, stellar leakage around the null, and instrument thermal emission).

Origin of Instability Noise

Figure 1: As the amplitude and phase errors vary with time, the residual photon leakage rate fluctuates and can mimic the presence of a planet.

In practice, it will not be possible to maintain the exact set of amplitudes and phases - vibrations and thermal drifts result in small path-length errors and time-variable aberrations. The null "floor" is degraded, and there is a time-variable leakage of stellar photons that can mimic a planet signal (Fig. 1b and c). This is known as "instability noise" (previously "systematic error" or "variability noise"), and it increases the integration time required. The analysis of instability noise is somewhat complex (Lay 2004).

Some components are removed by phase-chopping. Others, such as the "amplitude-phase cross terms" and the "co-phasing error," are not removed and result in photon leakage. The analysis shows that a null depth of about 1 part in 100,000 is generally sufficient to control the level of photon noise from the stellar leakage, but that a null depth of about 1 part in 1,000,000 is needed to prevent instability noise from becoming the dominant source of noise. A null of 1 part in 1,000,000 requires rms path control to within ~1.5 nm, and rms amplitude control of ~0.1%. It is therefore instability noise, not photon noise that drives the performance of the instrument.

Table 1 lists the mechanisms responsible for amplitude and phase errors, along with their spectral dependence and temporal nature. For example, vibrations in the optical path difference (OPD) result in a phase error that scales as the inverse of wavelength and are inherently time varying. Beam shear also gives a time-varying amplitude error (less light is coupled to the detector) but is approximately independent of wavelength. Mismatches in dispersion, birefringence, and reflectivities between the collector beams may have a more complex spectral dependence, denoted by f (l), but are inherently static. These static effects can be compensated by an adaptive nuller (Peters et al. 2006) or similar device. The dynamic terms combine via amplitude-phase and co-phasing mechanisms to form the time-variable instability noise. Other instability mechanisms, such as an unbalanced chop, may also be important, but will have a similar spectral dependence. The important conclusion is that at any given time the instability noise has a spectral signature that varies slowly with wavelength. This slow dependence with wavelength forms the basis of the first mitigation strategy, described in the following section.

Stretched X-Array and Spectral Filtering

In the stretched X-array and spectral filtering approach, we use the spectral dependence as the means to distinguish between the planet signal and the instability noise. To make this effective, the array must be stretched significantly with respect to previous designs. The X-array is the natural choice, since it can be stretched along its long dimension while preserving the short nulling baselines needed to minimize the stellar leakage. In principle, all instability noise can be eliminated. The nulling requirement can be relaxed from 1 part in 1,000,000 to 1 part in 100,000, while at the same time the sensitivity is improved and the angular resolution of the array is significantly increased. The technique is described below. A more complete analysis can be found in Lay (2006).

Figure 2. X-ray configurations.

Figure 2a shows the stretched X-array geometry, consisting of four collectors located on the corners of a rectangle and a central combiner, observing a star normal to the plane of the page. The aspect ratio has been stretched to 6:1, with 35-m nulling baselines. The corresponding instrument response at a wavelength of 10 µm, projected onto the plane of the sky, is shown in Fig. 2b. This is the "chopped" response - the phasing of the array is switched rapidly between two states, and the difference is taken. The star is located in the middle of the central, vertical null stripe (mid-grey); on either side the response has both positive (white) and negative (black) regions. A single planet is shown with a radial offset of 50 mas from the star. The detected photon rate from the planet as a function of rotation azimuth of the array is shown in Fig. 2c. The circular symbol gives the photon rate for the rotation angle shown in Fig. 2b. The peaks and valleys of Fig. 2c correspond to the white and black parts of the response along the circular locus in Fig. 2b. As the wavelength is increased from 10 µm, the instrument response of Fig. 2b is scaled about the center, with increased spacing between the peaks and valleys of the response. The photon rate from the planet is also changed according to its spectral distribution. Figure 2d combines these effects to show how the planet photon rate depends on both the wavelength (or optical frequency) and the array rotation azimuth. The example is based on a planet with a 265-K black-body spectrum, which has a substantially higher photon rate at 20 µm compared to 6 µm. A horizontal section through this distribution at a wavelength of 10 µm gives the profile shown in Fig. 2c. The wavelength-azimuth plot is a convenient representation of the data obtained from spectral channels of TPF-I as the array is rotated.

Figure 3. Vertical cuts.

In addition to the planet signal, there are two distinct classes of noise. The photon (shot) noise is shown in Fig. 2e, and is proportional to the square root of the overall photon rate. Important contributors are the local and exozodiacal backgrounds and stellar leakage. The instrument instability noise is shown in Fig. 2f. We assume that the full spectral range of 6 to 20 µm has been split for practical reasons into two bands for nulling: 6-10 µm and 10-20 µm. (It is difficult to cover the full range with one set of glasses and single-mode spatial filters.) Over each of these bands the instability noise at any instant is represented by a low-order polynomial series in the optical frequency multiplied by the stellar spectrum, according to Eq. 1. The coefficients vary randomly with time as the instabilities (path length, tilt, etc.) evolve. In Fig. 2f we show instability noise that is random from one azimuth to the next and from one band to the other (i.e., having a white noise spectrum). In practice the spectrum is not exactly white, and there will be some correlation, both with azimuth and between the spectral bands, but we will not rely on this correlation for the analysis presented here. In general, the instability noise increases at high optical frequencies / short wavelengths. The smooth variation with wavelength, coupled with the white-noise spectrum in azimuth, result in the distinctive vertical striping seen in the plot.

Figure 3a shows vertical cuts through the wavelength-azimuth plot at an azimuth of 30 degrees, and depicts both the planet signal and an example of instability noise. The instability noise is a smooth, slowly varying function within the two halves of the spectrum, whereas the planet signal oscillates with wavelength. We exploit this difference to remove the instability noise. Removing a low-order polynomial fit from each half of the spectrum gives the curves shown in Fig. 3b. In each case, the instability noise signature has been almost totally removed, and the signal remains largely intact, although somewhat modified. The impact of the fitting on the planet signature is predictable and can be corrected. Some of the planet signal is removed by the fitting process (which impacts the sensitivity), and it is this that motivates the need for a stretched array. As the array size is reduced, there are fewer oscillations of the planet signal across the spectrum, and more of the planet signal is removed by the low-order fit. The 6:1 aspect ratio described here is a compromise between the array size and the amount of planet signal that is lost.

With the effective removal of the instability noise, it is possible to relax the required null depth. Table 2 lists the SNR obtained for both broad-band detection and ozone spectroscopy, relative to the SNR with a 1 part in 1,000,000 null depth. In the absence of instability noise, the SNR is determined by photon noise, with principal contributions from stellar size leakage, local zodiacal dust, and stellar-null floor leakage. Only the stellar-null floor leakage depends on the null depth. Relaxing the null depth to 1 part in 100,000 has only a small impact on the SNR. At a null depth of 1 part in 10,000 the stellar-null floor leakage is becoming the dominant source of photon noise. But even with this relaxation by a factor of 100, the mission is still viable, albeit with reduced sensitivity.

A significant added benefit of the stretched array is that the angular resolution is improved by a factor of ~3.