Signal to noise ratio

To calculate the method's sensitivity in a more general way than done by Labeyrie (1995), we first follow his derivation to the point where he calculates the signal-to-noise ratio.
The different photon distribution from the star and the planet defines the signal-to-noise ratio SNR, and according to the central limit theorem:

$$nP_*(0)[1-P_0(0)]=SNR\sqrt{nP_*(0)}$$ (1)

where P_*(k_*) and P₀(k₀) are the probabilities to detect k_* and k₀ photons originating respectively from the star and from the planet, per pixel in a short exposure, and where n is the total number of short exposures. Replacing the values of P_*(0) and P₀(0) in Eq. (1), we obtain:

$$SNR=(1-e^{-\overline{k_0}})\sqrt{n\over 1+\overline{k_*}}$$ (2)

The calculation of signal to noise ratio given in Labeyrie 1995 assumed $\overline{k_*}\gg 1$, which is unnecessarily restrictive, and unrealistic in some of the cases of interest. As suggested by one of us (RR), a more general analysis can be made under the assumption that $\overline{k_0}\ll 1$ and $\overline{k_*}\gg\overline{k_0}$. Equation (2) then becomes:

$$SNR\approx \overline{k_0}\sqrt{n\over 1+\overline{k_*}}$$ (3)

if j is the number of pixels per speckles, thus:

$$(SNR)^2\approx {{n\over j}(j\overline{k_0})^2\over j+j\overline{k_*}}={n'\overline{K_0}^2\over j+\overline{K_*}}$$ (4)

Where $\overline {K_0}$ and $\overline {K_*}$ are the number of photons per speckle in a short exposure, respectively for the planet and the star.
The variables used in [Labeyrie 1995] were:
n'=${T\over t}$, $\;\;{\overline {K_*} \over \overline{K_0}}$=${R\over G}\;\;$ and $\;\;G\overline {K_*}=tN_*$
where T is the total integration time, t the short exposure time, R the star/planet brightness ratio, G the gain of the adaptive optics or the ratio between the Airy peak and the halo of speckles, N_* the total number of photons per second detected from the star.
It provides a new expression for the SNR:

The sampling j should be fine enough to exploit the darkest parts of the dark speckles, for a given threshold of detection $\epsilon$, linking the performance of the adaptive optics (G) and the brightness ratio (R). The intensity across a dark speckle may be coarsely modelled as a cosine function of the position $\rho$ by:

$$I(\rho)=I_h\left(1-\cos{2\pi\rho\over \sqrt j}\right)$$ (5)

where I_h is the mean intensity and $\sqrt j$ the speckle size. The intersection between $I(\rho)$ and the line $y=\epsilon I_h$ gives s, the size of the pixel over which the light is integrated. It limits the minimal measurable intensity $\epsilon I_h$ (with $0<\epsilon \ll 1$). A detailed calculation gives $s\approx 1.27\sqrt {\epsilon}$${\lambda\over D}$.
We can assume that $\epsilon=G/R$.Indeed, if I₀ is the intensity of the Airy peak, $\epsilon=$${\epsilon I_h\over I_h}$=${\epsilon I_h\over I_0}{I_0\over I_h}$=${G\over R}$
We also assume in the following that a planet can be seen if its own intensity is higher than $\epsilon I_h$.
Now we are able to calculate a value of j:

$$j={\left(\lambda /D\right)^2\over s^2}={\left(\lambda /D\rig... ...over {1.27^2\epsilon \left(\lambda /D\right)^2}}=0.62{R\over G}$$ (6)

 

Figure: Shape of a dark speckle according to Eq. (6). If I_h is the mean intensity, integration on a pixel of size s yields an intensity $\epsilon I_h$.

Recent numerical simulations ([Boccaletti 1998]) have shown that R/G should not exceed 10³ to retain a reasonable value of the sampling parameter j.
The recorded level is I_h, but with dark speckles this residual level decreases to $\epsilon I_h$, where $\epsilon$ depends mainly on the adaptive optics performance.
With the value of j obtained, the SNR expression becomes:

$$SNR={N_*\over R}\sqrt{tT\over 0.62{R\over G}+{tN_*\over G}}$$ (7)

Solving for R, leads to a third-degree equation:

$$0.62R^3+tN_*R^2={GtTN_*^2\over (SNR)^2}$$ (8)

The Cardan method gives a single positive solution from which we derive a final expression for the integration time.

$$T=\left({SNR\over N_*}\right)^2{R\over Gt}(0.62R^2+tN_*R)$$ (9)

With the following values : R=10⁹; D=8m; G=10⁶; SNR=5; m_v=2.5; q=0.2; $\Delta\lambda=100nm$ and t=20ms, we find 2.7 hours of integration, i.e. $50\%$ more than the result given by Labeyrie.
Angel's discussion (1994) of the long-exposure method leads him to the following expression:

$$R={G\over SNR}\sqrt{T\over \Delta t_{opt}}$$ (10)

where $\Delta t_{opt}$ is the optimum short exposure time. Using again the same values, Eq.(11) gives a limiting brightness ratio of 1.4.10⁸ and the typical brightness ratio of 10⁹ would be reached in 140 hours. The long-exposure approach would in principle be more sensitive if the adaptive optics and shadow pattern compensation could be made extremely good. It is however less sensitive with current levels of adaptive performance.
In fact, the difference between the dark-speckle and direct-long-exposure methods is more subtle, and both work in different regimes. A critical value of the photon stellar flux (N_*c) can be easily calculated by equalizing Eq.(10) and Eq.(11) for the same value of the total integration time (T). It leads to a second degree equation in N_*, with a single positive root:

$$N_{*c}={G+\sqrt{G^2+4\times 0.62GR}\over 2t}$$ (11)

Therefore, if the photon flux is above N_*c, the dark-speckle method is more efficient than the long exposure and, below this limit the direct imaging is better. N_*c strongly depends on the adaptive optics. Let us take a numerical exemple. If the goal is to reach a 10⁹ brightness ratio with a gain of 10⁶ and 20ms exposures, the photon flux must be higher than 1.27.10⁹ photons/s, which can be achieved with a large aperture and wide bandwidth (9.3.10⁹ph/s for m_v=0, D=2.4m, $\Delta\lambda=100nm$ and 20% efficiency). However, for a space telescope with adaptive optics, the speckle lifetime is under control with values of the order of 1s for exemple. Thus, N_*c is decreased to 2.54.10⁷ photons/s, which is less restrictive. However, the telescope should not be so large as to provide a partially resolved image of the star, since it would fill-in the dark speckles.