Erratum for: A&A 561, A118 (2014), DOI: 10.1051/0004-6361/201322665

In Sect. 3.3 of Kellerer (2014) I suggested detector read-out times below the coherence time of photons. If one assumes, instead, read-out times not less than the coherence time, Δt = λ²/(c Δλ), the spontaneous photons exceed, even for a very small field-of-view, the number of stimulated photons per incoming photon.

With the notation employed in  Kellerer (2014) the mean number of cloned photons per incoming photon is: (1)where I is the number of excited atoms, σ is the cross-section of excited atoms and S is the aperture- and amplifier-area. A field of angular diameter θ = 2.44 λ/D – where D is the aperture diameter – corresponds to the Airy disc up to its first minimum. Within the read-out time Δt = λ²/ (c Δλ), equal to the photon coherence time, this “diffraction area” receives a mean number of spontaneous photons: (2)A is the spontaneous emission rate.

From these relations one obtains the average fluence ratio of spontaneous and stimulated photons on the diffraction area: (3)

in line with calculations by Prasad (1994) and his conclusions that the spontaneous emissions dominate the stimulated ones. On the other hand, on average 0.64 N cloned photons end up on the central standard deviation range of diameter 1/3 of the Airy disc. This area is 9 times smaller than the Airy disc considered above. Thus the ratio of spontaneous to stimulated fluence is not 7.3 but merely 7.3/(9 × 0.64) ~ 1.3 in this region around the centre of the cloned photons.

The spontaneous photons will prevent a large improvement of resolution as long as our set-up lacks a stage to recognize events where the stimulated emissions dominate. Such a stage is in principle possible, see notably the probabilistic noiseless amplification processes discussed by Duan & Guo (1998), Ralph & Lund (2009). The main message of my article remains: it is fundamentally possible to improve the resolution of a telescope beyond the diffraction limit at the price of sensitivity, i.e. it is possible to trade sensitivity against resolution. The set-up that I have suggested will be incomplete unless it is given a suitable heralding stage.

References

© ESO, 2015