The usability of the optical parametric amplification of light for high-angular-resolution imaging and fast astrometry

The usability of the optical parametric amplification of light for high-angular-resolution... Abstract High-angular-resolution imaging is crucial for many applications in modern astronomy and astrophysics. The fundamental diffraction limit constrains the resolving power of both ground-based and spaceborne telescopes. The recent idea of a quantum telescope based on the optical parametric amplification (OPA) of light aims to bypass this limit for the imaging of extended sources by an order of magnitude or more. We present an updated scheme of an OPA-based device and a more accurate model of the signal amplification by such a device. The semiclassical model that we present predicts that the noise in such a system will form so-called light speckles as a result of light interference in the optical path. Based on this model, we analysed the efficiency of OPA in increasing the angular resolution of the imaging of extended targets and the precise localization of a distant point source. According to our new model, OPA offers a gain in resolved imaging in comparison to classical optics. For a given time-span, we found that OPA can be more efficient in localizing a single distant point source than classical telescopes. instrumentation: high angular resolution, techniques: high angular resolution, telescopes, astrometry 1 INTRODUCTION It is not possible to obtain precise resolved images of the vast majority of astrophysical targets of interest with current technology because of the large distances involved. The current technological limits of the diameter of the telescope primary mirror are ∼40 m on the ground (Group 2006; Liske 2011) and ∼6.5 m in space (Mather 2012). This is still many orders of magnitude below the size that would allow the resolved imaging of celestial objects such as most stars and the central regions of galaxies. Therefore, the brightness, temporal brightness variations (light-curves) and temperature are the primary sources of information about the physical sizes of observed objects. The highest angular resolution available today in the optical range of wavelength (which in astronomy is usually considered to be 400–1100 nm) is ∼0.1 arcsec, while the largest angular diameter of a star as seen from the Earth is 0.05 arcsec (50 mas, Betelgeuse; Uitenbroek et al. 1998). Other desired targets are even smaller angularly; for example, the predicted size of the prominent features of the event horizon of the black hole Sagittarius A* at the Galactic Centre is ∼30 μas (micro arcsecond) (Falcke, Melia & Agol 2000). Although existing optical interferometers achieve a resolution in imaging of up to 200 μ (CHARA Interferometer), their imaging capability is severely limited and they can operate only on very bright targets (Eisenhauer et al. 2011). The upcoming Event Horizon Telescope is expected to provide polarimetric imaging with 25-μ resolution at 1.3 mm in 2018 (Chael et al. 2016). However, no existing telescope project is aimed at improving the diffraction limit of the instrument (Kulkarni 2016). In order to increase the angular resolution, it is desirable to read the information on the exact direction of every photon, the photons will contribute to the image, before most of this information is disturbed as a result of diffraction in the optical system. This could be done if there existed a method to produce exact copies of each photon before the diffraction occurs, and to register the copies so that the statistical analysis can be performed (online or offline). In other words, it is desirable to amplify the signal before registering it. The optical parametric amplification (OPA) of light is a process in which an input signal (photon) is amplified and almost exact copies of it are created by pumping energy into the amplifying medium (e.g. a laser crystal). Caves (1982) showed that this process is too noisy to enable an immediate increase of the resolution of imaging beyond the diffraction limit. In recent years, however, this limitation has been generalized: it was found that it is possible to decrease the registered noise amount significantly by the use of a trigger signal (Ralph & Lund 2009; Barbieri et al. 2011). This idea of an OPA-based device for astronomical observations, often referred to as a quantum telescope (QT), was later further investigated by Kellerer (2014b) and Kurek et al. (2016). In Section 2 of this paper we discuss the idea of a telescope based on the OPA in detail. One drawback of all the models of a QT discussed so far is that they are based on very simplified models of noise. However, the correct modelling of noise in such systems, and especially the statistical distribution of the noise, will be crucial for a proper assessment of the efficiency of such a device. As a step to address this issue, in this article we present a semiclassical model of the OPA as applied for astronomical imaging, which, in particular, provides a more accurate prediction of the noise than previous models (Kellerer 2014a,b; Kurek et al. 2016). The paper is organized as follows. In Section 2.1 we describe the idea of using OPA to increase the angular resolution in astronomy. In Section 2.2 we present previous research on the use of OPA in astronomy. In Section 2.3 we describe the influence of intrinsic OPA noise on the resolution gain. In Section 3 we introduce the semiclassical model of the OPA, which we then apply to characterizing the efficiency. Section 4 describes our numerical simulations. We use two methods of signal analysis: (i) estimation of the position of the centroid of the signal, and (ii) estimation of the position of maxima of speckles. For the first method, the results of the simulations of the efficiency in the angular resolution and localization of a distant point source are presented in, respectively, Sections 5.1 and 5.2. For the latter method, the results are presented in Sections 6.1 and 6.2. We conclude in Section 7. 2 QUANTUM TELESCOPE 2.1 Concept The idea of a QT was recently presented in Kellerer (2014a,b). The aim of the idea is to increase the resolving power of existing and future telescopes by introducing new optical elements into the optical path. In such a telescope, each photon inbound from a celestial extended source is first detected by a quantum non-demolition (QND) device, which registers the time of the photon's arrival. QND is a type of measurement of a quantum system in which the observable is negligibly changed; see Guerlin et al. (2007) for further details. Most importantly, the photon is not absorbed by QND detection. As the arrival time is known, it is possible to turn on the detector only for a specified time interval, during which the photon's arrival is expected. After detection, the photon passes through a pumped amplifying medium (e.g. a Beta barium BOrate crystal, BBO) and is cloned by parametric amplification (Scarani et al. 2005; Lamas-Linares et al. 2002; Barbieri et al. 2011). The photon from the astronomical target and its clones (stimulated emission) are from this moment indistinguishable and treated as one photon cloud that is registered by a fast 2D coincidence detector, for example on an ICCD (intensified charge-coupled device, Lampton 1981) or EMCCD (electron multiplying charge coupled device, also known as a low light level CCD, L3CCD) camera (Tulloch & Dhillon 2011). Unfortunately, the amplifier produces unavoidable spontaneous emission, which is required by the uncertainty principle (Stenholm 1986). Although the very fast electronic gating of the detector (preferably not less than the coherence time of photons: Δt = λ2/(c Δλ)) prevents the system from registering too many spontaneous-emission photons, the final image is still contaminated by significant noise. The centroid position of the photon cloud is computed offline and passed as a count into a final high-resolution image. The entire process is illustrated in Fig. 1, which shows a toy-model of a QT. The process is repeated for every photon detected by the QND device, and in this way a high-resolution image is constructed, photon by photon, during a sufficiently long exposure. Figure 1. View largeDownload slide A general quantum telescope scheme. Elements are not to scale. Figure 1. View largeDownload slide A general quantum telescope scheme. Elements are not to scale. 2.2 Model The first QT setup proposed in Kellerer (2014b) was based on a QND device with an optical resonator (cavity). However, this would have caused a large amount of diffraction, preventing this QT setup from increasing the angular resolution. In Kurek et al. (2016), we suggested using a cavity-free QND device according to the scheme proposed by Xia et al. (2016), which in addition is much easier to use than the ‘classical’ QND device proposed by Haroche et al. (1999). In the cavity-free version, the trajectory of a photon is not affected, and the introduced diffraction is negligible because a 1D waveguide embedded with atoms is used. In such a 1D waveguide (a hollow-core photonic crystal fibre), the light is well confined (Benisty et al. 1999) in the 1D space as a guided mode (Keyu Xia, private communication). Furthermore, as we discuss below, a QT cannot be placed in the plane conjugated to the pupil; that is, a QT cannot be implemented as a small device located after the mirror, as proposed by Kellerer & Ribak (2016). The basic theory of quantum physics and optics indicates that the position of a photon, whose wavefunction is reduced to the size of the telescope's aperture, cannot be retrieved with an accuracy exceeding the diffraction limit. Therefore, in order to extend the resolution of a telescope beyond this limit, the QND device and cloning stages need to be placed before the aperture (i.e. before the diffraction takes place), as shown in Fig. 2. In this setup, the clones preserve the information of the original photon. The cloud of clones, although diffracted by the telescope aperture, still centres around the original position. In other words, each clone is diffracted and its position is randomly changed, but the cloud of clones preserves its centroid. Importantly, the detection in the QND device does not introduce diffraction, because only the arrival time of the photon is detected, not its position. In contrast, locating the QND device and cloning stages after the telescope mirror cannot increase the resolution. The cloud of clones, as a whole, is randomly shifted, because the primary photon experiences diffraction. A QT placed behind the aperture works in fact as a simple photon intensifier (like a multichannel plate), which can never increase the resolution (and usually degrades it). The design according to our proposed scheme (e.g. Fig. 2) is obviously more complicated technologically, because it requires a large amplifying medium and a large QND detector. QND devices are still in their initial stage of development, but, to the best of our knowledge, there is no premise for a size limitation on such a device. Figure 2. View largeDownload slide Scheme of a quantum telescope. The quantum addition to the telescope is placed in front of the mirror. Figure 2. View largeDownload slide Scheme of a quantum telescope. The quantum addition to the telescope is placed in front of the mirror. 2.3 Noise According to Kellerer (2015) and Prasad (1994), a QT has a large intrinsic noise: the amplification reduces the signal-to-noise ratio (SNR) to ≲1/7.3. This was supposed to be the lowest fundamental limit of noise in this design, originating from the necessity of opening the wells of the detector for at least the photon coherence time – which is long enough for the photon to be absorbed (Kellerer 2015; Prasad 1994). Using the matched filtering (MF) approach,1 we proved (Kurek et al. 2016) that even such a high level of noise does not prevent the QT from achieving a good performance, as long as a sufficient number of clones is provided. The QT toy-model was demonstrated to perform better than a classic telescope (CT) of the same mirror size, if on average ≥56 clones are provided for every photon from space (Kurek et al. 2016). It is obviously desirable to amplify a signal that is usually weak in astronomy. However, deterministic optical amplification (frequently referred to as quantum cloning of light) is necessarily noisy (Caves 1982): any amplifier working independently on the phase of the input signal introduces noise because of the uncertainty principle and no-cloning theorem (Ferreyrol et al. 2010). The desired perfect amplification for a QT would be noiseless, namely $$\left|\alpha \right\rangle \rightarrow \left|\sqrt{G}\alpha \right\rangle$$, where |α〉 is a coherent state of light and G is the amplification gain. Such a procedure is also in principle possible probabilistically, if a much more complex amplifier were used. A few working schemes have already been demonstrated: Ralph & Lund (2009); Zavatta, Fiurášek & Bellini (2011); Marek & Filip (2010). However, all of them are very difficult to realize using present technology, mainly because they use multiple quantum-scissors — a device able to generate any desired superposition of the vacuum and one-photon states (Pegg, Phillips & Barnett 1998). Nevertheless, the functioning of such quantum-scissors was recently demonstrated by Xiang et al. (2010) and Ferreyrol et al. (2010). 3 QT – A SEMICLASSICAL MODEL In the previous analysis (Kurek et al. 2016), the noise was simulated by a Poissonian process and the clone count per frame was also distributed according to the Poissonian process. This is a default choice for rare events; see Fox (2006, chapter 5) for a derivation. The clone cloud was assumed to have a Gaussian shape. Below we present a more accurate 2D model of a QT that fully includes spatial intensity correlations in the image plane. Let us consider first a point source located on the instrument axis. The normalized mode function in the pupil plane is defined over a circle Π(ρ) of radius D/2. The circle has the following form:   \begin{eqnarray} \Pi ({\boldsymbol{\rho }}) =\left\lbrace \begin{array}{@{}l@{\quad }l@{}}1/\sqrt{\pi D^2/4}, & \quad \mbox{if }|{\boldsymbol{\rho }}| < D/2 \\ 0 & \quad \mbox{if }|{\boldsymbol{\rho }}| \ge D/2, \end{array}\right. \end{eqnarray} (1)where D is the telescope mirror diameter. We use the polar coordinates ρ (radial) and ϕ (angular) in the pupil plane. The centre of the polar coordinate system is equivalent to the centre of the pupil plane. In order to account for the noise generated by the amplifier (‘cloning device’) in the pupil plane, it is necessary to complement the signal mode $$\tilde{u}_{00}(\rho ,\phi )$$ with orthonormal modes that together form a complete set over the pupil area. The signal can be decomposed by orthogonal modes in many different ways, so that we can choose any orthogonal set of functions for the description of the modes. Because we assume that the pupil is circular, the natural choice of orthogonal functions is the Zernike polynomials. A convenient choice (Tatulli 2013) is given by the Zernike modes $$\tilde{u}_{nm}({\boldsymbol{\rho },\phi })$$, where n = 0, 1, 2, … and m = −n, −n + 2, …, n − 2, n. Their explicit form can be found in, for example, Noll (1976). In the image plane, contributions from individual modes are given by the corresponding 2D Fourier transforms, which explicitly read:   \begin{eqnarray} u_{nm}(\tilde{\rho },\tilde{\phi })=(-1)^{n}\sqrt{n+1}\frac{J_{n+1}(2\pi \tilde{\rho })}{\pi \tilde{\rho }}\! \left\lbrace \begin{array}{@{}l@{\quad }l@{}}\begin{array}{l}\sqrt{2}\cos (\mid m\mid \tilde{\phi })\,\text{ if }\,m>0\\ \sqrt{2}\sin (\mid m\mid \tilde{\phi })\,\text{ if }\,m<0\\ 1\text{ if }\,m=0, \end{array} \end{array}\right. {}\nonumber \\ \end{eqnarray} (2)where Jn is the nth Bessel function, $$\tilde{\rho }$$ and $$\tilde{\phi }$$ are Fourier-transformed variables, and $$\tilde{\rho }$$ is expressed in units of pixels. We assume that the amplifier has a uniform gain factor G for all spatial modes in the pupil plane; that is, the amplitude of an individual mode is multiplied by a factor $$\sqrt{G}$$, whereas the mean number of spontaneously generated photons in that mode is G − 1 (Haus 2000, chapter 9.3). We use a semiclassical model to describe light fluctuations, assigning to each mode a dimensionless complex amplitude αnm whose squared absolute value |αnm|2 characterizes the average photon number in that mode. In this paper, a semiclassical model means that the model is based on the first quantization theory: we treat the light beam as a classical electromagnetic wave and the amplification of the signal as a classical process, and we assume the semiclassical theory of photodetection. We assume the the source is at an infinite distance. This assumption allows us to treat the signal as coherent light with unit amplitude. A given cloning medium is able to produce clones only in a very narrow wavelength span. So we can filter the light and pass only a given narrow wavelength span. Therefore, we can assume that the photon distribution is not thermal, and as a consequence the difference in their wavelengths is very small. Single-mode thermal light is governed by the Bose–Einstein distribution (Fox 2006). Because every emission in the cloning medium is independent of the others (Haus 2000, chapter 9), the singular emission is described by a singular mode. The multimode distribution of the light is thus a result of the convolution of a large number of single-mode distributions. Therefore, after the amplification process the amplitude is described by a probability distribution:   \begin{equation} p(\tilde{\alpha }_{00}) = \frac{1}{\pi (G-1)} \exp \left( - \frac{|\tilde{\alpha }_{00}|^2}{G-1} \right), \end{equation} (3)where $$\tilde{\alpha }_{00}=\alpha _{00}-\sqrt{G}$$. The amplitudes αnm for all other noise modes unm with n > 0 are characterized by a Gaussian thermal distribution:   \begin{equation} p(\alpha _{nm}) = \frac{1}{\pi (G-1)} \exp \left( - \frac{|\alpha _{nm} |^2}{G-1} \right), \quad n=1,2,\ldots \ . \end{equation} (4)We use equations (3) and (4) to find the modulus of the random coefficients αnm. Because equation (4) depends only on |αnm|, all the values of the phases of the αnm coefficients are treated the same by equation (4). As a result, the values of the phases of the αnm coefficients are generated from the uniform distribution from the interval 〈0, 2π). If we know the value of the phase and the modulus of the $$\tilde{\alpha }_{00}$$ coefficient, we know the α00 coefficient because $$\alpha _{00}=\tilde{\alpha }_{00}+\sqrt{G}$$. 4 SIMULATIONS We performed our numerical simulations using an iterative code prepared in the Matlab package. In order to test the usability of multimode OPA for high-angular-resolution astronomical imaging, we implemented the model described above. Note that equation (2) is undefined when both arguments are equal to zero. At the same time, point (0, 0) is important for the accuracy of the simulations, because in a perfect situation (no noise) the centroid of the signal should be located there. According to Popowicz, Kurek & Filus (2013), biharmonic interpolation is the most efficient of all the interpolation methods used frequently in astronomy. Therefore, we fitted a biharmonic surface to the region surrounding the pixel corresponding to  $$(\tilde{\rho },\tilde{\phi })$$ = 0 and copied the appropriate value into this pixel. Because previously the value of this pixel was not defined, there is no possibility of estimating the goodness of such a fit at this pixel. The computed value is larger by 0.1985 per cent than the median of the eight nearest surrounding pixels (see Fig. 3). Figure 3. View largeDownload slide Electromagnetic wave density around u1, 1(0, 0). Left: a pixel of undefined value is visible in the middle. There are four highest points in this image: (5, 6), (7, 6), (6, 5) and (6, 7). Middle and right: the value estimated from the fitting was pasted to this pixel. There is only one highest point: (6, 6). Figure 3. View largeDownload slide Electromagnetic wave density around u1, 1(0, 0). Left: a pixel of undefined value is visible in the middle. There are four highest points in this image: (5, 6), (7, 6), (6, 5) and (6, 7). Middle and right: the value estimated from the fitting was pasted to this pixel. There is only one highest point: (6, 6). In order to simulate an individual image frame, a set of complex amplitudes αnm was randomly chosen according to equations (3) and (4). For this realization, the intensity distribution in the image plane is given by   \begin{equation} I(\tilde{\rho },\tilde{\phi }) = \Biggl | \sum _{{n=0} }^N \sum _m^n \alpha _{nm} u_{nm} (\tilde{\rho },\tilde{\phi }) \Biggr |^2, \end{equation} (5)where, as previously, m = −n, −n + 2, …, n − 2,  n. In practice, the summation over n is truncated to a cut-off value N = 100 for which the included modes are effectively complete in the image area of interest. Numerically, this means that the sum $$\sum _{nm} |u_{nm} (\tilde{\rho },\tilde{\phi })|^2$$ with n = 0, 1, …, N should be sufficiently close to one over the respective area. In our simulations, we found that the deviation from one for the above sum is $${\lesssim }10^{-11}$$. In the last step, the image plane is discretized into pixels, and the number of photons in an individual pixel is chosen according to a Poissonian distribution with the mean equal to $$I(\tilde{\rho },\tilde{\phi })$$ times the pixel area. The above procedure describes a simulation for a point source. For a non-point incoherent light source, it is necessary to choose a single point emitting radiation in the source plane in the first step and then to follow the procedure described above. The only modification is that the mode functions $$u_{nm}(\tilde{\rho },\tilde{\phi })$$ in the image plane introduced in equation (2) should be displaced correspondingly to the location of the source. Care should be taken that the displaced mode functions also satisfy the effective completeness condition within the imaging area. Let r = |αnm| denote the amplitude of the single mode. Then equations (3) and (4) can be rewritten in polar coordinates as   \begin{equation} p(r,\theta )=\frac{1}{\pi (G-1)}\exp \left(\frac{-r^2+2\sqrt{G}r\cos \theta -G}{G-1}\right) \end{equation} (6)and   \begin{equation} p(r,\theta )=\frac{1}{\pi (G-1)}\exp \left(-\frac{r^2}{G-1}\right), \end{equation} (7)respectively. We can marginalize the above equations over the θ coordinate. In consequence, we obtain   \begin{equation} p(r)=\frac{2}{(G-1)}r\exp \left(\frac{r^2+G}{1-G}\right)I_0\left(\frac{2\sqrt{G}r}{G-1}\right) \end{equation} (8)and   \begin{equation} p(r)=\frac{2}{(G-1)}r\exp \left(\frac{r^2}{1-G}\right), \end{equation} (9)respectively. Ii(x) is the modified Bessel function of the first kind. Let r0 denote the amplitude of the signal. The probability that the amplitude of the single mode of the noise is greater than r0 is given by the formula   \begin{equation} F(r_0)=\int _{r_0}^\infty p(r)\,{\rm d}r=\exp \left(\frac{r_0^2}{1-G}\right), \end{equation} (10)where p(r) is given by equation (9). The probability that the amplitude of a mode of the noise is greater than the amplitude of the signal is   \begin{equation} E = \int _0^\infty F(r_0) p(r_0)\,{\rm d}r_0 = \frac{1}{2}\exp \left(\frac{G}{2(1-G)}\right), \end{equation} (11)where p(r0) is given by equation (8). From the above equation we can obtain the probability that the amplitude of any mode of noise is greater than the amplitude of the signal:   \begin{eqnarray} P &=& \sum _{i=1}^N(-1)^{i+1}{{N}\atopwithdelims (){i}}E^i \nonumber \\ &=& \sum _{i=1}^N(-1)^{i+1}{{N}\atopwithdelims (){i}}\frac{1}{2^i}\exp \left(\frac{i G}{2(1-G)}\right)\nonumber \\ &=& 1-\left(1-\frac{1}{2}\exp \left(\frac{G}{2(1-G)}\right)\right)^N, \end{eqnarray} (12)where N is the number of modes of the noise. 5 RESULTS – CENTROID SEARCH METHOD Fig. 4 presents an example outcome of the simulations: (a) the density of an electromagnetic wave in the image plane, (b) the surface distribution of the photons, and (c) an image as it would be registered by a modern EMCCD camera2, which introduces excess (multiplication) noise. The result depicted in Fig. 4(c) demonstrates that the camera efficiency (registration error) is not an essential issue here, as the image degradation after adding the registration error (SSIM3 = 0.7060) is insignificant in terms of distinguishing the clump structures. For a comparison with an experiment, we can look at fig. 2 from Mosset et al. (2004), which presents experimentally acquired light speckles that are very similar to our simulated images (however, Mosset et al. (2004) present the result of a single-mode light experiment, which is governed by the Bose–Einstein distribution, and therefore some deviation would be justified). The noise has a clumpy structure, and the signal looks like one of the clumps, because the modes of both the signal and the clumpy structure are described by the same expression : equation (2). The only difference between the signal and the clumps is given by the probability distribution of the amplitude of modes (see equations 3 and 4). Such clumps in the laser optics are called speckles (Dainty 1970, 1975). Figure 4. View largeDownload slide (a) Exemplary realization of the density of the electromagnetic wave in the image plane for an OPA gain G = 15; (b) the density of photon-events in the image plane; (c) added simulated noise from the high-end EMCCD / L3CCD camera image (gain = 100, saturation level = 105 e-). A realistic detector introduces an additional uncertainty, mainly photon multiplication noise and clock-inducted charge (CIC) noise, which is omitted here, because in modern EMCCDs it is negligible. The scale depends on the distance of the image plane from the focus, and therefore we do not associate it with a metric unit here. The black pixel inside the inset denotes the centre of the image. Figure 4. View largeDownload slide (a) Exemplary realization of the density of the electromagnetic wave in the image plane for an OPA gain G = 15; (b) the density of photon-events in the image plane; (c) added simulated noise from the high-end EMCCD / L3CCD camera image (gain = 100, saturation level = 105 e-). A realistic detector introduces an additional uncertainty, mainly photon multiplication noise and clock-inducted charge (CIC) noise, which is omitted here, because in modern EMCCDs it is negligible. The scale depends on the distance of the image plane from the focus, and therefore we do not associate it with a metric unit here. The black pixel inside the inset denotes the centre of the image. In the MF-based signal analysis used in Kurek et al. (2016), the simulated image was first convolved with the Gaussian profile (σ = 10), and then the centroid (the centre of the signal photon cloud) was obtained from the position of the maximum value of such a filtered image. As in the outcome of equation (2) in the present model, both the signal and the noise have the form of on average identical clumps; there is no obvious reason MF would distinguish the signal from the noise, as was the case in the previous models. The only way to recover the signal is to assume that in some realizations the signal clump may be stronger than the noise clumps. This is a consequence of the probability distributions of the amplitude of the signal (equation 3) and clumps (equation 4). Therefore, we looked for the highest point in the image, as was done in Kurek et al. (2016). In the example realization of the density of the electromagnetic wave in the image plane depicted in Fig. 4, the position of the signal is clearly easy to recover, but in most cases the signal was not localized correctly. The probability that the amplitude of any mode of noise is greater than the amplitude of the signal is given by formula (12). In Fig. 5, the gain parameter G = 15. Fig. 5(a) presents an example realization, where the signal is supposed to be located in the middle, but it is not visible. Fig. 5(b) presents a covariance matrix of this event. Fig. 5(c) presents the square root of the covariance matrix to reduce the contrast. Figs 5(d), (e) and (f) present the analogical information, but for the average of 20 consecutive events. While it is not guaranteed that it will be possible to localize the signal in any particular event, it is always possible in the average of a large enough number of events. FWHM (full width at half-maximum) sizes and intensities of the noise clumps are on average very similar across the considered field of view, which is depicted by the diagonal in Figs 5(b), (c), (e) and (f). Because the modes of the signal and the clumps are given by the same formula (equation 2), the shape and size of noise clumps are indistinguishable from the signal of interest (clone cloud), which is best shown by the diagonal in Fig. 5(f). Equation (12) describes statistically how often the signal steps out above the noise as a function of the gain G. Figure 5. View largeDownload slide (a) One simulated G = 15 amplification event, as it would be registered by an EMCCD camera (the signal is supposed to be in the middle, but, in contrast to the event depicted in Fig. 4, is not visible in this particular realization); (b) the covariance matrix of this event; (c) the square root of the covariance matrix (to reduce the contrast); (d) the average of 20 simulated events; (e) averaged covariances of 20 simulated events; (f) square root of the averaged covariance matrix. The black cross-hair denotes the centre of the average of covariance matrices. Figure 5. View largeDownload slide (a) One simulated G = 15 amplification event, as it would be registered by an EMCCD camera (the signal is supposed to be in the middle, but, in contrast to the event depicted in Fig. 4, is not visible in this particular realization); (b) the covariance matrix of this event; (c) the square root of the covariance matrix (to reduce the contrast); (d) the average of 20 simulated events; (e) averaged covariances of 20 simulated events; (f) square root of the averaged covariance matrix. The black cross-hair denotes the centre of the average of covariance matrices. 5.1 Angular resolution – extended source In order to compare the efficiency of a CT and a QT in the high-angular-resolution imaging of extended sources, we compared the ‘pencil size’ of each one. To simulate results that would be obtained by a CT, we simulated the classical optics image formation of a distant point source. In this process, an Airy pattern is successively drawn by incoming photons. In the case of the CT, we distributed the signal according to the squared absolute values of u0, 0, which is in fact a diffraction (Airy) pattern of the assumed optical system. In this distribution, the average Euclidean distance of a count from the centre of the image (depicted in Fig. 6 with red circles) was 42.97 pixels. For reference, the mean Euclidean distance of all pixels in the frame from the central pixel, computed from the equation   \begin{equation} d_{\rm all} =\frac{\sum _{x=1}^{1001}\sum _{y=1}^{1001} \sqrt{(x-501)^2+(y-501)^2} }{ 1001^2}, \end{equation} (13)was 383.98 pixels. Figure 6. View largeDownload slide The example results of the centroid estimation error overplotted on the log intensity-scaled diffraction pattern. Left: unprocessed frame. Middle: blurred frame. Right: matched-filtering frame. The average error for a quantum telescope is in blue; that for a classic telescope is in red. Figure 6. View largeDownload slide The example results of the centroid estimation error overplotted on the log intensity-scaled diffraction pattern. Left: unprocessed frame. Middle: blurred frame. Right: matched-filtering frame. The average error for a quantum telescope is in blue; that for a classic telescope is in red. In the case of the QT, after each amplification event we computed the Euclidean distance of the highest peak from the known true source position. We performed three groups of tests. In the first one, we did not modify the frame in any way. In the second one, we convolved the frame with a Gaussian spanning 5 × 5 pixels and with σ = 0.5. This was done in order to blur the image slightly and thereby to reduce the influence of the shot noise on the centroid estimation. In the third test, we performed MF on the averaged frame using the diffraction pattern as a known template. This is in essence equivalent to test 2, but instead of an arbitrary assumed Gaussian, we used a diffraction pattern in the convolution. The results varied depending on the amplification gain G and the signal analysis method. An example depicting the procedure is presented in Fig. 6. In this example, the mean localization error (‘pencil size’) in the case of the QT varies from 313.32 (σ = 190.31) pixels in both the unprocessed and blurred frames to 277.47 (σ = 180.34) in the MF frame. All test were performed using 1001 × 1001 frames. Fig. 7 presents the results for all three tests. For every tested G, the centroid error for the QT is many times below the one for the CT (∼43 pixels). But it is still better than the average distance from the centre of a random pixel within the frame (equation 13), so, in the case of all G, statistically there is a weak correlation between the highest peak and the true position of the source. Figure 7. View largeDownload slide Centroid estimation error of the quantum telescope in high-angular-resolution extended source imaging. All three types of tests are presented. Each point represents a mean of 1000 iterations. The average random pixel error is overplotted by the dashed line. Figure 7. View largeDownload slide Centroid estimation error of the quantum telescope in high-angular-resolution extended source imaging. All three types of tests are presented. Each point represents a mean of 1000 iterations. The average random pixel error is overplotted by the dashed line. In sum, in the framework of the model presented here, the noise has a correlated space-dependent structure, which was not assumed in previous models (Kurek et al. 2016). Therefore, there is no obvious way to localize the signal, because in most cases the highest peak appears anywhere in the frame at a position that is too weakly correlated with the position of the signal (see equation 12), so at this step the procedure fails. In order to obtain any gain in the resolution, the signal localization error for every event would have to be on average lower than the Airy disc FWHM. This result implies that the use of the centroid search method for signal analysis from OPA in astronomy does not allow for a resolution gain in the imaging of extended sources. 5.2 Localization of a distant point source Another possible use of QTs are astrometric measurements of the position of a distant point source. As before, we performed our simulations using images of 1001 × 1001 pixels. The count of photons constituting an undersampled Airy pattern was equal to the amplification event count in the QT. For the CT, we computed the dot product centroid of the signal and the Euclidean distance of the centroid to the centre of the diffraction pattern of the source, the position of which was estimated. In the case of the QT, we searched for the highest point in the averaged set of images of the outcome of the amplification process degraded by the shot noise. As in the case of the extended source, we used three methods of signal analysis for the QT: an analysis of (a) raw data, (b) blurred data and (c) match-filtered data. According to our tests, for the same number of events, the localization error of the QT is on average lower than that of the classical optics. Fig. 8 presents exemplary result for G = 2, 10, 50, 200 and 1000 of the centroid estimation error as a function of photon (in the case of a CT) and amplification event (in the case of a QT) counts. The tests based on processed QT data (blurred and match-filtered) show a lower error than the test based on raw data. For relatively small numbers of events (10–20) the supremacy of the QT over the CT is unstable, but it stabilizes for larger iteration counts. The exact result (Fig. 8) is a non-linear function of: the gain G (weak dependence), the photon/event count (strong dependence) and the method of centroid computation (weak dependence), but, as a rule of thumb, the Euclidean distance error tends to be ∼3 times smaller with the use of OPA, if the event number exceeds 10–20. Figure 8. View largeDownload slide Comparison of centroid errors produced by classic and quantum telescopes for the localization of a point source. Green dots denote the efficiency of classical optics. For each G, the results for the classic telescope were computed anew. Circles, crosses and diamonds denote the quantum telescope/optical parametric amplification efficiency. Figure 8. View largeDownload slide Comparison of centroid errors produced by classic and quantum telescopes for the localization of a point source. Green dots denote the efficiency of classical optics. For each G, the results for the classic telescope were computed anew. Circles, crosses and diamonds denote the quantum telescope/optical parametric amplification efficiency. The superiority of OPA probably originates from the fact that, beginning from small numbers of events, the signal in an averaged frame is more intense than the noise clumps, and thus provides accurate information on the position of the source. At the same time, in the case of the CT the Airy pattern is still not fully drawn, so the shot noise still significantly increases the localization error. 6 RESULTS – SPECKLE MAXIMA METHOD Another method of signal analysis that we tested is based on the localization of the maxima of all the speckles. After acquiring each frame of 1001 × 1001 pixels, we performed MF to remove the effects of the shot noise. In the next step, we analysed the frame using a sliding window of 3 × 3 pixels. In order to avoid boundary-condition problems, we ran the sliding window from pixel (2, 2) and ended it at (end-1, end-1). This is justified, because the signal is expected in the middle of the frame. We checked if the pixel in the middle of the window had a higher value than any other pixel in the window. If this was the case, the pixel was marked as a peak of a speckle. In this way we localized the maxima of all the speckles present in the frame (Fig. 9). We tested this approach by carefully investigating the localized positions, and no evidently false detections were found. This implies that the MF is able to cancel out efficiently the effects of shot noise. The only questionable detections are localized near the edges of the frame, but they do not influence the results, because, as noted, the signal is expected in the middle of the frame – which is far enough from the edges (e.g. Fig. 9, left). Figure 9. View largeDownload slide Localization of the maxima of the speckles. Top row: 1 iteration; bottom row: stacks of 20 iterations. Left column: OPA output; middle: localized maxima of the speckles (0-1 logic); right: localized maxima of the speckles – every maximum is represented by its value. Note that the maxima of elongated speckles are also localized correctly. Gain G = 10. Figure 9. View largeDownload slide Localization of the maxima of the speckles. Top row: 1 iteration; bottom row: stacks of 20 iterations. Left column: OPA output; middle: localized maxima of the speckles (0-1 logic); right: localized maxima of the speckles – every maximum is represented by its value. Note that the maxima of elongated speckles are also localized correctly. Gain G = 10. 6.1 Angular resolution – extended source After stacking a large enough number of localization results, a Gaussian-like shape is visible in the middle of the stack (Fig. 9, bottom right). In the current method, we assume that it originates from the signal clumps and that it can be fitted by a 2D Gaussian surface. In order to ensure stable fit results, we do this using a stack of 2000 frames. We measured the FWHM of such a fitted Gaussian and compared it with the FWHM of u0, 0. The results for a series of gain G values are shown in Fig. 10. The FWHM obtained with the use of OPA (mean 19.45 pixels, σ = 1.27 pixels) is significantly narrower than the one obtained with the use of classical optics (50.45 pixels). This confirms that OPA is able to produce a gain in the resolution of 2D imaging. Figure 10. View largeDownload slide Top: comparison of the FWHM for quantum and classic telescopes. Bottom: goodness of fit for a quantum telescope (r2). Figure 10. View largeDownload slide Top: comparison of the FWHM for quantum and classic telescopes. Bottom: goodness of fit for a quantum telescope (r2). 6.2 Localization of a distant point source As in Section 5.2, we checked the efficiency for 1∼1000 events and a series of gain G values. After acquiring localization frames (e.g. Fig. 9, bottom right), we stacked them and performed MF using u0,0 as a convolution kernel. We computed the Euclidean distance from the highest point of the convolution results to the middle of the frame and legitimized this distance as a point-source localization error. The results are presented in Fig. 11. It can be seen that OPA is able to achieve the precision limit of such a test more than an order of magnitude faster than a CT. Figure 11. View largeDownload slide Comparison of the localization error of a point source using the speckle maxima method. Green dots denote the efficiency of classic optics. For each G, the results for the classic telescope were computed anew. Blue dots denote the quantum telescope/optical parametric amplification efficiency. Figure 11. View largeDownload slide Comparison of the localization error of a point source using the speckle maxima method. Green dots denote the efficiency of classic optics. For each G, the results for the classic telescope were computed anew. Blue dots denote the quantum telescope/optical parametric amplification efficiency. 7 CONCLUSIONS Our semi-classical model, which is more accurate then previously considered models, shows that the intrinsic parametric amplification noise tends to form random clumps when passing the telescope optics. This effect was not included in the previous models of a QT (Kellerer 2014a,b, 2015; Kurek et al. 2016; Kellerer & Ribak 2016) based on the first quantization. However, such clumps, called light speckles, have been shown experimentally to exist (Mosset et al. 2004). The presence of these clumps means that it is difficult to distinguish the signal from the noise and implies that a different signal analysis procedure should be applied. Further modelling and analyses of such noise in greater detail are needed, because an understanding of this issue is essential for the practical realization of a QT. According to our new results, the centroid search approach will not produce any considerable resolution gain. However, the approach based on localizing the maxima of the speckles – the second signal analysis method we tested – offers about a threefold improvement over classical optics. Moreover, both signal analysis methods were shown to be more efficient in the localization of a point source than classical optics, given the same photon count per measurement. Here again the speckle maxima method is superior to the centroid search approach. In principle, it is possible to improve the performance further with the use of noiseless amplification instead of the OPA. However, this approach has never been tested in astronomy (Ralph & Lund 2009; Zavatta et al. 2011; Marek & Filip 2010). Recently, other techniques competitive to OPA have emerged for overcoming the diffraction limit in far-field imaging (Chrostowski et al. 2017). These possibilities are currently under investigation by our group. Acknowledgements We thank our colleague, Dr R. Demkowicz-Dobrzanski, for useful and inspiring discussions, and Dr A. Popowicz for valuable suggestions concerning signal analysis. We also thank the anonymous referee for useful and constructive comments which led to significant improvements in the manuscript. Funding was provided by the Polish National Science Centre grant no. 2016/21/N/ST9/00375, the European Commission FP7 projects SIQS (grant agreement no. 600645) and PhoQuS@UW (grant agreement no. 316244) co-financed by the Polish Ministry of Science and Higher Education (MNSW: 71501E-338/M/2017) and the Foundation for Polish Science TEAM project ‘Quantum Optical Communication Systems’ co-financed by the European Union under the European Regional Development Fund. Footnotes 1 MF is an efficient detector of the known template in a noisy environment and is widely used in radars or sonars, where weak, well-defined reflected signals have to be detected (Helstrom 1968; Woodward 1953). 2 For a review of EMCCD image formation, see section 2 in Tulloch & Dhillon (2011). 3 The structural similarity index is a measure of image similarity. Its value ranges from −1 to 1 (identical images). For details, see Wang et al. (2004). 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The usability of the optical parametric amplification of light for high-angular-resolution imaging and fast astrometry

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Abstract

Abstract High-angular-resolution imaging is crucial for many applications in modern astronomy and astrophysics. The fundamental diffraction limit constrains the resolving power of both ground-based and spaceborne telescopes. The recent idea of a quantum telescope based on the optical parametric amplification (OPA) of light aims to bypass this limit for the imaging of extended sources by an order of magnitude or more. We present an updated scheme of an OPA-based device and a more accurate model of the signal amplification by such a device. The semiclassical model that we present predicts that the noise in such a system will form so-called light speckles as a result of light interference in the optical path. Based on this model, we analysed the efficiency of OPA in increasing the angular resolution of the imaging of extended targets and the precise localization of a distant point source. According to our new model, OPA offers a gain in resolved imaging in comparison to classical optics. For a given time-span, we found that OPA can be more efficient in localizing a single distant point source than classical telescopes. instrumentation: high angular resolution, techniques: high angular resolution, telescopes, astrometry 1 INTRODUCTION It is not possible to obtain precise resolved images of the vast majority of astrophysical targets of interest with current technology because of the large distances involved. The current technological limits of the diameter of the telescope primary mirror are ∼40 m on the ground (Group 2006; Liske 2011) and ∼6.5 m in space (Mather 2012). This is still many orders of magnitude below the size that would allow the resolved imaging of celestial objects such as most stars and the central regions of galaxies. Therefore, the brightness, temporal brightness variations (light-curves) and temperature are the primary sources of information about the physical sizes of observed objects. The highest angular resolution available today in the optical range of wavelength (which in astronomy is usually considered to be 400–1100 nm) is ∼0.1 arcsec, while the largest angular diameter of a star as seen from the Earth is 0.05 arcsec (50 mas, Betelgeuse; Uitenbroek et al. 1998). Other desired targets are even smaller angularly; for example, the predicted size of the prominent features of the event horizon of the black hole Sagittarius A* at the Galactic Centre is ∼30 μas (micro arcsecond) (Falcke, Melia & Agol 2000). Although existing optical interferometers achieve a resolution in imaging of up to 200 μ (CHARA Interferometer), their imaging capability is severely limited and they can operate only on very bright targets (Eisenhauer et al. 2011). The upcoming Event Horizon Telescope is expected to provide polarimetric imaging with 25-μ resolution at 1.3 mm in 2018 (Chael et al. 2016). However, no existing telescope project is aimed at improving the diffraction limit of the instrument (Kulkarni 2016). In order to increase the angular resolution, it is desirable to read the information on the exact direction of every photon, the photons will contribute to the image, before most of this information is disturbed as a result of diffraction in the optical system. This could be done if there existed a method to produce exact copies of each photon before the diffraction occurs, and to register the copies so that the statistical analysis can be performed (online or offline). In other words, it is desirable to amplify the signal before registering it. The optical parametric amplification (OPA) of light is a process in which an input signal (photon) is amplified and almost exact copies of it are created by pumping energy into the amplifying medium (e.g. a laser crystal). Caves (1982) showed that this process is too noisy to enable an immediate increase of the resolution of imaging beyond the diffraction limit. In recent years, however, this limitation has been generalized: it was found that it is possible to decrease the registered noise amount significantly by the use of a trigger signal (Ralph & Lund 2009; Barbieri et al. 2011). This idea of an OPA-based device for astronomical observations, often referred to as a quantum telescope (QT), was later further investigated by Kellerer (2014b) and Kurek et al. (2016). In Section 2 of this paper we discuss the idea of a telescope based on the OPA in detail. One drawback of all the models of a QT discussed so far is that they are based on very simplified models of noise. However, the correct modelling of noise in such systems, and especially the statistical distribution of the noise, will be crucial for a proper assessment of the efficiency of such a device. As a step to address this issue, in this article we present a semiclassical model of the OPA as applied for astronomical imaging, which, in particular, provides a more accurate prediction of the noise than previous models (Kellerer 2014a,b; Kurek et al. 2016). The paper is organized as follows. In Section 2.1 we describe the idea of using OPA to increase the angular resolution in astronomy. In Section 2.2 we present previous research on the use of OPA in astronomy. In Section 2.3 we describe the influence of intrinsic OPA noise on the resolution gain. In Section 3 we introduce the semiclassical model of the OPA, which we then apply to characterizing the efficiency. Section 4 describes our numerical simulations. We use two methods of signal analysis: (i) estimation of the position of the centroid of the signal, and (ii) estimation of the position of maxima of speckles. For the first method, the results of the simulations of the efficiency in the angular resolution and localization of a distant point source are presented in, respectively, Sections 5.1 and 5.2. For the latter method, the results are presented in Sections 6.1 and 6.2. We conclude in Section 7. 2 QUANTUM TELESCOPE 2.1 Concept The idea of a QT was recently presented in Kellerer (2014a,b). The aim of the idea is to increase the resolving power of existing and future telescopes by introducing new optical elements into the optical path. In such a telescope, each photon inbound from a celestial extended source is first detected by a quantum non-demolition (QND) device, which registers the time of the photon's arrival. QND is a type of measurement of a quantum system in which the observable is negligibly changed; see Guerlin et al. (2007) for further details. Most importantly, the photon is not absorbed by QND detection. As the arrival time is known, it is possible to turn on the detector only for a specified time interval, during which the photon's arrival is expected. After detection, the photon passes through a pumped amplifying medium (e.g. a Beta barium BOrate crystal, BBO) and is cloned by parametric amplification (Scarani et al. 2005; Lamas-Linares et al. 2002; Barbieri et al. 2011). The photon from the astronomical target and its clones (stimulated emission) are from this moment indistinguishable and treated as one photon cloud that is registered by a fast 2D coincidence detector, for example on an ICCD (intensified charge-coupled device, Lampton 1981) or EMCCD (electron multiplying charge coupled device, also known as a low light level CCD, L3CCD) camera (Tulloch & Dhillon 2011). Unfortunately, the amplifier produces unavoidable spontaneous emission, which is required by the uncertainty principle (Stenholm 1986). Although the very fast electronic gating of the detector (preferably not less than the coherence time of photons: Δt = λ2/(c Δλ)) prevents the system from registering too many spontaneous-emission photons, the final image is still contaminated by significant noise. The centroid position of the photon cloud is computed offline and passed as a count into a final high-resolution image. The entire process is illustrated in Fig. 1, which shows a toy-model of a QT. The process is repeated for every photon detected by the QND device, and in this way a high-resolution image is constructed, photon by photon, during a sufficiently long exposure. Figure 1. View largeDownload slide A general quantum telescope scheme. Elements are not to scale. Figure 1. View largeDownload slide A general quantum telescope scheme. Elements are not to scale. 2.2 Model The first QT setup proposed in Kellerer (2014b) was based on a QND device with an optical resonator (cavity). However, this would have caused a large amount of diffraction, preventing this QT setup from increasing the angular resolution. In Kurek et al. (2016), we suggested using a cavity-free QND device according to the scheme proposed by Xia et al. (2016), which in addition is much easier to use than the ‘classical’ QND device proposed by Haroche et al. (1999). In the cavity-free version, the trajectory of a photon is not affected, and the introduced diffraction is negligible because a 1D waveguide embedded with atoms is used. In such a 1D waveguide (a hollow-core photonic crystal fibre), the light is well confined (Benisty et al. 1999) in the 1D space as a guided mode (Keyu Xia, private communication). Furthermore, as we discuss below, a QT cannot be placed in the plane conjugated to the pupil; that is, a QT cannot be implemented as a small device located after the mirror, as proposed by Kellerer & Ribak (2016). The basic theory of quantum physics and optics indicates that the position of a photon, whose wavefunction is reduced to the size of the telescope's aperture, cannot be retrieved with an accuracy exceeding the diffraction limit. Therefore, in order to extend the resolution of a telescope beyond this limit, the QND device and cloning stages need to be placed before the aperture (i.e. before the diffraction takes place), as shown in Fig. 2. In this setup, the clones preserve the information of the original photon. The cloud of clones, although diffracted by the telescope aperture, still centres around the original position. In other words, each clone is diffracted and its position is randomly changed, but the cloud of clones preserves its centroid. Importantly, the detection in the QND device does not introduce diffraction, because only the arrival time of the photon is detected, not its position. In contrast, locating the QND device and cloning stages after the telescope mirror cannot increase the resolution. The cloud of clones, as a whole, is randomly shifted, because the primary photon experiences diffraction. A QT placed behind the aperture works in fact as a simple photon intensifier (like a multichannel plate), which can never increase the resolution (and usually degrades it). The design according to our proposed scheme (e.g. Fig. 2) is obviously more complicated technologically, because it requires a large amplifying medium and a large QND detector. QND devices are still in their initial stage of development, but, to the best of our knowledge, there is no premise for a size limitation on such a device. Figure 2. View largeDownload slide Scheme of a quantum telescope. The quantum addition to the telescope is placed in front of the mirror. Figure 2. View largeDownload slide Scheme of a quantum telescope. The quantum addition to the telescope is placed in front of the mirror. 2.3 Noise According to Kellerer (2015) and Prasad (1994), a QT has a large intrinsic noise: the amplification reduces the signal-to-noise ratio (SNR) to ≲1/7.3. This was supposed to be the lowest fundamental limit of noise in this design, originating from the necessity of opening the wells of the detector for at least the photon coherence time – which is long enough for the photon to be absorbed (Kellerer 2015; Prasad 1994). Using the matched filtering (MF) approach,1 we proved (Kurek et al. 2016) that even such a high level of noise does not prevent the QT from achieving a good performance, as long as a sufficient number of clones is provided. The QT toy-model was demonstrated to perform better than a classic telescope (CT) of the same mirror size, if on average ≥56 clones are provided for every photon from space (Kurek et al. 2016). It is obviously desirable to amplify a signal that is usually weak in astronomy. However, deterministic optical amplification (frequently referred to as quantum cloning of light) is necessarily noisy (Caves 1982): any amplifier working independently on the phase of the input signal introduces noise because of the uncertainty principle and no-cloning theorem (Ferreyrol et al. 2010). The desired perfect amplification for a QT would be noiseless, namely $$\left|\alpha \right\rangle \rightarrow \left|\sqrt{G}\alpha \right\rangle$$, where |α〉 is a coherent state of light and G is the amplification gain. Such a procedure is also in principle possible probabilistically, if a much more complex amplifier were used. A few working schemes have already been demonstrated: Ralph & Lund (2009); Zavatta, Fiurášek & Bellini (2011); Marek & Filip (2010). However, all of them are very difficult to realize using present technology, mainly because they use multiple quantum-scissors — a device able to generate any desired superposition of the vacuum and one-photon states (Pegg, Phillips & Barnett 1998). Nevertheless, the functioning of such quantum-scissors was recently demonstrated by Xiang et al. (2010) and Ferreyrol et al. (2010). 3 QT – A SEMICLASSICAL MODEL In the previous analysis (Kurek et al. 2016), the noise was simulated by a Poissonian process and the clone count per frame was also distributed according to the Poissonian process. This is a default choice for rare events; see Fox (2006, chapter 5) for a derivation. The clone cloud was assumed to have a Gaussian shape. Below we present a more accurate 2D model of a QT that fully includes spatial intensity correlations in the image plane. Let us consider first a point source located on the instrument axis. The normalized mode function in the pupil plane is defined over a circle Π(ρ) of radius D/2. The circle has the following form:   \begin{eqnarray} \Pi ({\boldsymbol{\rho }}) =\left\lbrace \begin{array}{@{}l@{\quad }l@{}}1/\sqrt{\pi D^2/4}, & \quad \mbox{if }|{\boldsymbol{\rho }}| < D/2 \\ 0 & \quad \mbox{if }|{\boldsymbol{\rho }}| \ge D/2, \end{array}\right. \end{eqnarray} (1)where D is the telescope mirror diameter. We use the polar coordinates ρ (radial) and ϕ (angular) in the pupil plane. The centre of the polar coordinate system is equivalent to the centre of the pupil plane. In order to account for the noise generated by the amplifier (‘cloning device’) in the pupil plane, it is necessary to complement the signal mode $$\tilde{u}_{00}(\rho ,\phi )$$ with orthonormal modes that together form a complete set over the pupil area. The signal can be decomposed by orthogonal modes in many different ways, so that we can choose any orthogonal set of functions for the description of the modes. Because we assume that the pupil is circular, the natural choice of orthogonal functions is the Zernike polynomials. A convenient choice (Tatulli 2013) is given by the Zernike modes $$\tilde{u}_{nm}({\boldsymbol{\rho },\phi })$$, where n = 0, 1, 2, … and m = −n, −n + 2, …, n − 2, n. Their explicit form can be found in, for example, Noll (1976). In the image plane, contributions from individual modes are given by the corresponding 2D Fourier transforms, which explicitly read:   \begin{eqnarray} u_{nm}(\tilde{\rho },\tilde{\phi })=(-1)^{n}\sqrt{n+1}\frac{J_{n+1}(2\pi \tilde{\rho })}{\pi \tilde{\rho }}\! \left\lbrace \begin{array}{@{}l@{\quad }l@{}}\begin{array}{l}\sqrt{2}\cos (\mid m\mid \tilde{\phi })\,\text{ if }\,m>0\\ \sqrt{2}\sin (\mid m\mid \tilde{\phi })\,\text{ if }\,m<0\\ 1\text{ if }\,m=0, \end{array} \end{array}\right. {}\nonumber \\ \end{eqnarray} (2)where Jn is the nth Bessel function, $$\tilde{\rho }$$ and $$\tilde{\phi }$$ are Fourier-transformed variables, and $$\tilde{\rho }$$ is expressed in units of pixels. We assume that the amplifier has a uniform gain factor G for all spatial modes in the pupil plane; that is, the amplitude of an individual mode is multiplied by a factor $$\sqrt{G}$$, whereas the mean number of spontaneously generated photons in that mode is G − 1 (Haus 2000, chapter 9.3). We use a semiclassical model to describe light fluctuations, assigning to each mode a dimensionless complex amplitude αnm whose squared absolute value |αnm|2 characterizes the average photon number in that mode. In this paper, a semiclassical model means that the model is based on the first quantization theory: we treat the light beam as a classical electromagnetic wave and the amplification of the signal as a classical process, and we assume the semiclassical theory of photodetection. We assume the the source is at an infinite distance. This assumption allows us to treat the signal as coherent light with unit amplitude. A given cloning medium is able to produce clones only in a very narrow wavelength span. So we can filter the light and pass only a given narrow wavelength span. Therefore, we can assume that the photon distribution is not thermal, and as a consequence the difference in their wavelengths is very small. Single-mode thermal light is governed by the Bose–Einstein distribution (Fox 2006). Because every emission in the cloning medium is independent of the others (Haus 2000, chapter 9), the singular emission is described by a singular mode. The multimode distribution of the light is thus a result of the convolution of a large number of single-mode distributions. Therefore, after the amplification process the amplitude is described by a probability distribution:   \begin{equation} p(\tilde{\alpha }_{00}) = \frac{1}{\pi (G-1)} \exp \left( - \frac{|\tilde{\alpha }_{00}|^2}{G-1} \right), \end{equation} (3)where $$\tilde{\alpha }_{00}=\alpha _{00}-\sqrt{G}$$. The amplitudes αnm for all other noise modes unm with n > 0 are characterized by a Gaussian thermal distribution:   \begin{equation} p(\alpha _{nm}) = \frac{1}{\pi (G-1)} \exp \left( - \frac{|\alpha _{nm} |^2}{G-1} \right), \quad n=1,2,\ldots \ . \end{equation} (4)We use equations (3) and (4) to find the modulus of the random coefficients αnm. Because equation (4) depends only on |αnm|, all the values of the phases of the αnm coefficients are treated the same by equation (4). As a result, the values of the phases of the αnm coefficients are generated from the uniform distribution from the interval 〈0, 2π). If we know the value of the phase and the modulus of the $$\tilde{\alpha }_{00}$$ coefficient, we know the α00 coefficient because $$\alpha _{00}=\tilde{\alpha }_{00}+\sqrt{G}$$. 4 SIMULATIONS We performed our numerical simulations using an iterative code prepared in the Matlab package. In order to test the usability of multimode OPA for high-angular-resolution astronomical imaging, we implemented the model described above. Note that equation (2) is undefined when both arguments are equal to zero. At the same time, point (0, 0) is important for the accuracy of the simulations, because in a perfect situation (no noise) the centroid of the signal should be located there. According to Popowicz, Kurek & Filus (2013), biharmonic interpolation is the most efficient of all the interpolation methods used frequently in astronomy. Therefore, we fitted a biharmonic surface to the region surrounding the pixel corresponding to  $$(\tilde{\rho },\tilde{\phi })$$ = 0 and copied the appropriate value into this pixel. Because previously the value of this pixel was not defined, there is no possibility of estimating the goodness of such a fit at this pixel. The computed value is larger by 0.1985 per cent than the median of the eight nearest surrounding pixels (see Fig. 3). Figure 3. View largeDownload slide Electromagnetic wave density around u1, 1(0, 0). Left: a pixel of undefined value is visible in the middle. There are four highest points in this image: (5, 6), (7, 6), (6, 5) and (6, 7). Middle and right: the value estimated from the fitting was pasted to this pixel. There is only one highest point: (6, 6). Figure 3. View largeDownload slide Electromagnetic wave density around u1, 1(0, 0). Left: a pixel of undefined value is visible in the middle. There are four highest points in this image: (5, 6), (7, 6), (6, 5) and (6, 7). Middle and right: the value estimated from the fitting was pasted to this pixel. There is only one highest point: (6, 6). In order to simulate an individual image frame, a set of complex amplitudes αnm was randomly chosen according to equations (3) and (4). For this realization, the intensity distribution in the image plane is given by   \begin{equation} I(\tilde{\rho },\tilde{\phi }) = \Biggl | \sum _{{n=0} }^N \sum _m^n \alpha _{nm} u_{nm} (\tilde{\rho },\tilde{\phi }) \Biggr |^2, \end{equation} (5)where, as previously, m = −n, −n + 2, …, n − 2,  n. In practice, the summation over n is truncated to a cut-off value N = 100 for which the included modes are effectively complete in the image area of interest. Numerically, this means that the sum $$\sum _{nm} |u_{nm} (\tilde{\rho },\tilde{\phi })|^2$$ with n = 0, 1, …, N should be sufficiently close to one over the respective area. In our simulations, we found that the deviation from one for the above sum is $${\lesssim }10^{-11}$$. In the last step, the image plane is discretized into pixels, and the number of photons in an individual pixel is chosen according to a Poissonian distribution with the mean equal to $$I(\tilde{\rho },\tilde{\phi })$$ times the pixel area. The above procedure describes a simulation for a point source. For a non-point incoherent light source, it is necessary to choose a single point emitting radiation in the source plane in the first step and then to follow the procedure described above. The only modification is that the mode functions $$u_{nm}(\tilde{\rho },\tilde{\phi })$$ in the image plane introduced in equation (2) should be displaced correspondingly to the location of the source. Care should be taken that the displaced mode functions also satisfy the effective completeness condition within the imaging area. Let r = |αnm| denote the amplitude of the single mode. Then equations (3) and (4) can be rewritten in polar coordinates as   \begin{equation} p(r,\theta )=\frac{1}{\pi (G-1)}\exp \left(\frac{-r^2+2\sqrt{G}r\cos \theta -G}{G-1}\right) \end{equation} (6)and   \begin{equation} p(r,\theta )=\frac{1}{\pi (G-1)}\exp \left(-\frac{r^2}{G-1}\right), \end{equation} (7)respectively. We can marginalize the above equations over the θ coordinate. In consequence, we obtain   \begin{equation} p(r)=\frac{2}{(G-1)}r\exp \left(\frac{r^2+G}{1-G}\right)I_0\left(\frac{2\sqrt{G}r}{G-1}\right) \end{equation} (8)and   \begin{equation} p(r)=\frac{2}{(G-1)}r\exp \left(\frac{r^2}{1-G}\right), \end{equation} (9)respectively. Ii(x) is the modified Bessel function of the first kind. Let r0 denote the amplitude of the signal. The probability that the amplitude of the single mode of the noise is greater than r0 is given by the formula   \begin{equation} F(r_0)=\int _{r_0}^\infty p(r)\,{\rm d}r=\exp \left(\frac{r_0^2}{1-G}\right), \end{equation} (10)where p(r) is given by equation (9). The probability that the amplitude of a mode of the noise is greater than the amplitude of the signal is   \begin{equation} E = \int _0^\infty F(r_0) p(r_0)\,{\rm d}r_0 = \frac{1}{2}\exp \left(\frac{G}{2(1-G)}\right), \end{equation} (11)where p(r0) is given by equation (8). From the above equation we can obtain the probability that the amplitude of any mode of noise is greater than the amplitude of the signal:   \begin{eqnarray} P &=& \sum _{i=1}^N(-1)^{i+1}{{N}\atopwithdelims (){i}}E^i \nonumber \\ &=& \sum _{i=1}^N(-1)^{i+1}{{N}\atopwithdelims (){i}}\frac{1}{2^i}\exp \left(\frac{i G}{2(1-G)}\right)\nonumber \\ &=& 1-\left(1-\frac{1}{2}\exp \left(\frac{G}{2(1-G)}\right)\right)^N, \end{eqnarray} (12)where N is the number of modes of the noise. 5 RESULTS – CENTROID SEARCH METHOD Fig. 4 presents an example outcome of the simulations: (a) the density of an electromagnetic wave in the image plane, (b) the surface distribution of the photons, and (c) an image as it would be registered by a modern EMCCD camera2, which introduces excess (multiplication) noise. The result depicted in Fig. 4(c) demonstrates that the camera efficiency (registration error) is not an essential issue here, as the image degradation after adding the registration error (SSIM3 = 0.7060) is insignificant in terms of distinguishing the clump structures. For a comparison with an experiment, we can look at fig. 2 from Mosset et al. (2004), which presents experimentally acquired light speckles that are very similar to our simulated images (however, Mosset et al. (2004) present the result of a single-mode light experiment, which is governed by the Bose–Einstein distribution, and therefore some deviation would be justified). The noise has a clumpy structure, and the signal looks like one of the clumps, because the modes of both the signal and the clumpy structure are described by the same expression : equation (2). The only difference between the signal and the clumps is given by the probability distribution of the amplitude of modes (see equations 3 and 4). Such clumps in the laser optics are called speckles (Dainty 1970, 1975). Figure 4. View largeDownload slide (a) Exemplary realization of the density of the electromagnetic wave in the image plane for an OPA gain G = 15; (b) the density of photon-events in the image plane; (c) added simulated noise from the high-end EMCCD / L3CCD camera image (gain = 100, saturation level = 105 e-). A realistic detector introduces an additional uncertainty, mainly photon multiplication noise and clock-inducted charge (CIC) noise, which is omitted here, because in modern EMCCDs it is negligible. The scale depends on the distance of the image plane from the focus, and therefore we do not associate it with a metric unit here. The black pixel inside the inset denotes the centre of the image. Figure 4. View largeDownload slide (a) Exemplary realization of the density of the electromagnetic wave in the image plane for an OPA gain G = 15; (b) the density of photon-events in the image plane; (c) added simulated noise from the high-end EMCCD / L3CCD camera image (gain = 100, saturation level = 105 e-). A realistic detector introduces an additional uncertainty, mainly photon multiplication noise and clock-inducted charge (CIC) noise, which is omitted here, because in modern EMCCDs it is negligible. The scale depends on the distance of the image plane from the focus, and therefore we do not associate it with a metric unit here. The black pixel inside the inset denotes the centre of the image. In the MF-based signal analysis used in Kurek et al. (2016), the simulated image was first convolved with the Gaussian profile (σ = 10), and then the centroid (the centre of the signal photon cloud) was obtained from the position of the maximum value of such a filtered image. As in the outcome of equation (2) in the present model, both the signal and the noise have the form of on average identical clumps; there is no obvious reason MF would distinguish the signal from the noise, as was the case in the previous models. The only way to recover the signal is to assume that in some realizations the signal clump may be stronger than the noise clumps. This is a consequence of the probability distributions of the amplitude of the signal (equation 3) and clumps (equation 4). Therefore, we looked for the highest point in the image, as was done in Kurek et al. (2016). In the example realization of the density of the electromagnetic wave in the image plane depicted in Fig. 4, the position of the signal is clearly easy to recover, but in most cases the signal was not localized correctly. The probability that the amplitude of any mode of noise is greater than the amplitude of the signal is given by formula (12). In Fig. 5, the gain parameter G = 15. Fig. 5(a) presents an example realization, where the signal is supposed to be located in the middle, but it is not visible. Fig. 5(b) presents a covariance matrix of this event. Fig. 5(c) presents the square root of the covariance matrix to reduce the contrast. Figs 5(d), (e) and (f) present the analogical information, but for the average of 20 consecutive events. While it is not guaranteed that it will be possible to localize the signal in any particular event, it is always possible in the average of a large enough number of events. FWHM (full width at half-maximum) sizes and intensities of the noise clumps are on average very similar across the considered field of view, which is depicted by the diagonal in Figs 5(b), (c), (e) and (f). Because the modes of the signal and the clumps are given by the same formula (equation 2), the shape and size of noise clumps are indistinguishable from the signal of interest (clone cloud), which is best shown by the diagonal in Fig. 5(f). Equation (12) describes statistically how often the signal steps out above the noise as a function of the gain G. Figure 5. View largeDownload slide (a) One simulated G = 15 amplification event, as it would be registered by an EMCCD camera (the signal is supposed to be in the middle, but, in contrast to the event depicted in Fig. 4, is not visible in this particular realization); (b) the covariance matrix of this event; (c) the square root of the covariance matrix (to reduce the contrast); (d) the average of 20 simulated events; (e) averaged covariances of 20 simulated events; (f) square root of the averaged covariance matrix. The black cross-hair denotes the centre of the average of covariance matrices. Figure 5. View largeDownload slide (a) One simulated G = 15 amplification event, as it would be registered by an EMCCD camera (the signal is supposed to be in the middle, but, in contrast to the event depicted in Fig. 4, is not visible in this particular realization); (b) the covariance matrix of this event; (c) the square root of the covariance matrix (to reduce the contrast); (d) the average of 20 simulated events; (e) averaged covariances of 20 simulated events; (f) square root of the averaged covariance matrix. The black cross-hair denotes the centre of the average of covariance matrices. 5.1 Angular resolution – extended source In order to compare the efficiency of a CT and a QT in the high-angular-resolution imaging of extended sources, we compared the ‘pencil size’ of each one. To simulate results that would be obtained by a CT, we simulated the classical optics image formation of a distant point source. In this process, an Airy pattern is successively drawn by incoming photons. In the case of the CT, we distributed the signal according to the squared absolute values of u0, 0, which is in fact a diffraction (Airy) pattern of the assumed optical system. In this distribution, the average Euclidean distance of a count from the centre of the image (depicted in Fig. 6 with red circles) was 42.97 pixels. For reference, the mean Euclidean distance of all pixels in the frame from the central pixel, computed from the equation   \begin{equation} d_{\rm all} =\frac{\sum _{x=1}^{1001}\sum _{y=1}^{1001} \sqrt{(x-501)^2+(y-501)^2} }{ 1001^2}, \end{equation} (13)was 383.98 pixels. Figure 6. View largeDownload slide The example results of the centroid estimation error overplotted on the log intensity-scaled diffraction pattern. Left: unprocessed frame. Middle: blurred frame. Right: matched-filtering frame. The average error for a quantum telescope is in blue; that for a classic telescope is in red. Figure 6. View largeDownload slide The example results of the centroid estimation error overplotted on the log intensity-scaled diffraction pattern. Left: unprocessed frame. Middle: blurred frame. Right: matched-filtering frame. The average error for a quantum telescope is in blue; that for a classic telescope is in red. In the case of the QT, after each amplification event we computed the Euclidean distance of the highest peak from the known true source position. We performed three groups of tests. In the first one, we did not modify the frame in any way. In the second one, we convolved the frame with a Gaussian spanning 5 × 5 pixels and with σ = 0.5. This was done in order to blur the image slightly and thereby to reduce the influence of the shot noise on the centroid estimation. In the third test, we performed MF on the averaged frame using the diffraction pattern as a known template. This is in essence equivalent to test 2, but instead of an arbitrary assumed Gaussian, we used a diffraction pattern in the convolution. The results varied depending on the amplification gain G and the signal analysis method. An example depicting the procedure is presented in Fig. 6. In this example, the mean localization error (‘pencil size’) in the case of the QT varies from 313.32 (σ = 190.31) pixels in both the unprocessed and blurred frames to 277.47 (σ = 180.34) in the MF frame. All test were performed using 1001 × 1001 frames. Fig. 7 presents the results for all three tests. For every tested G, the centroid error for the QT is many times below the one for the CT (∼43 pixels). But it is still better than the average distance from the centre of a random pixel within the frame (equation 13), so, in the case of all G, statistically there is a weak correlation between the highest peak and the true position of the source. Figure 7. View largeDownload slide Centroid estimation error of the quantum telescope in high-angular-resolution extended source imaging. All three types of tests are presented. Each point represents a mean of 1000 iterations. The average random pixel error is overplotted by the dashed line. Figure 7. View largeDownload slide Centroid estimation error of the quantum telescope in high-angular-resolution extended source imaging. All three types of tests are presented. Each point represents a mean of 1000 iterations. The average random pixel error is overplotted by the dashed line. In sum, in the framework of the model presented here, the noise has a correlated space-dependent structure, which was not assumed in previous models (Kurek et al. 2016). Therefore, there is no obvious way to localize the signal, because in most cases the highest peak appears anywhere in the frame at a position that is too weakly correlated with the position of the signal (see equation 12), so at this step the procedure fails. In order to obtain any gain in the resolution, the signal localization error for every event would have to be on average lower than the Airy disc FWHM. This result implies that the use of the centroid search method for signal analysis from OPA in astronomy does not allow for a resolution gain in the imaging of extended sources. 5.2 Localization of a distant point source Another possible use of QTs are astrometric measurements of the position of a distant point source. As before, we performed our simulations using images of 1001 × 1001 pixels. The count of photons constituting an undersampled Airy pattern was equal to the amplification event count in the QT. For the CT, we computed the dot product centroid of the signal and the Euclidean distance of the centroid to the centre of the diffraction pattern of the source, the position of which was estimated. In the case of the QT, we searched for the highest point in the averaged set of images of the outcome of the amplification process degraded by the shot noise. As in the case of the extended source, we used three methods of signal analysis for the QT: an analysis of (a) raw data, (b) blurred data and (c) match-filtered data. According to our tests, for the same number of events, the localization error of the QT is on average lower than that of the classical optics. Fig. 8 presents exemplary result for G = 2, 10, 50, 200 and 1000 of the centroid estimation error as a function of photon (in the case of a CT) and amplification event (in the case of a QT) counts. The tests based on processed QT data (blurred and match-filtered) show a lower error than the test based on raw data. For relatively small numbers of events (10–20) the supremacy of the QT over the CT is unstable, but it stabilizes for larger iteration counts. The exact result (Fig. 8) is a non-linear function of: the gain G (weak dependence), the photon/event count (strong dependence) and the method of centroid computation (weak dependence), but, as a rule of thumb, the Euclidean distance error tends to be ∼3 times smaller with the use of OPA, if the event number exceeds 10–20. Figure 8. View largeDownload slide Comparison of centroid errors produced by classic and quantum telescopes for the localization of a point source. Green dots denote the efficiency of classical optics. For each G, the results for the classic telescope were computed anew. Circles, crosses and diamonds denote the quantum telescope/optical parametric amplification efficiency. Figure 8. View largeDownload slide Comparison of centroid errors produced by classic and quantum telescopes for the localization of a point source. Green dots denote the efficiency of classical optics. For each G, the results for the classic telescope were computed anew. Circles, crosses and diamonds denote the quantum telescope/optical parametric amplification efficiency. The superiority of OPA probably originates from the fact that, beginning from small numbers of events, the signal in an averaged frame is more intense than the noise clumps, and thus provides accurate information on the position of the source. At the same time, in the case of the CT the Airy pattern is still not fully drawn, so the shot noise still significantly increases the localization error. 6 RESULTS – SPECKLE MAXIMA METHOD Another method of signal analysis that we tested is based on the localization of the maxima of all the speckles. After acquiring each frame of 1001 × 1001 pixels, we performed MF to remove the effects of the shot noise. In the next step, we analysed the frame using a sliding window of 3 × 3 pixels. In order to avoid boundary-condition problems, we ran the sliding window from pixel (2, 2) and ended it at (end-1, end-1). This is justified, because the signal is expected in the middle of the frame. We checked if the pixel in the middle of the window had a higher value than any other pixel in the window. If this was the case, the pixel was marked as a peak of a speckle. In this way we localized the maxima of all the speckles present in the frame (Fig. 9). We tested this approach by carefully investigating the localized positions, and no evidently false detections were found. This implies that the MF is able to cancel out efficiently the effects of shot noise. The only questionable detections are localized near the edges of the frame, but they do not influence the results, because, as noted, the signal is expected in the middle of the frame – which is far enough from the edges (e.g. Fig. 9, left). Figure 9. View largeDownload slide Localization of the maxima of the speckles. Top row: 1 iteration; bottom row: stacks of 20 iterations. Left column: OPA output; middle: localized maxima of the speckles (0-1 logic); right: localized maxima of the speckles – every maximum is represented by its value. Note that the maxima of elongated speckles are also localized correctly. Gain G = 10. Figure 9. View largeDownload slide Localization of the maxima of the speckles. Top row: 1 iteration; bottom row: stacks of 20 iterations. Left column: OPA output; middle: localized maxima of the speckles (0-1 logic); right: localized maxima of the speckles – every maximum is represented by its value. Note that the maxima of elongated speckles are also localized correctly. Gain G = 10. 6.1 Angular resolution – extended source After stacking a large enough number of localization results, a Gaussian-like shape is visible in the middle of the stack (Fig. 9, bottom right). In the current method, we assume that it originates from the signal clumps and that it can be fitted by a 2D Gaussian surface. In order to ensure stable fit results, we do this using a stack of 2000 frames. We measured the FWHM of such a fitted Gaussian and compared it with the FWHM of u0, 0. The results for a series of gain G values are shown in Fig. 10. The FWHM obtained with the use of OPA (mean 19.45 pixels, σ = 1.27 pixels) is significantly narrower than the one obtained with the use of classical optics (50.45 pixels). This confirms that OPA is able to produce a gain in the resolution of 2D imaging. Figure 10. View largeDownload slide Top: comparison of the FWHM for quantum and classic telescopes. Bottom: goodness of fit for a quantum telescope (r2). Figure 10. View largeDownload slide Top: comparison of the FWHM for quantum and classic telescopes. Bottom: goodness of fit for a quantum telescope (r2). 6.2 Localization of a distant point source As in Section 5.2, we checked the efficiency for 1∼1000 events and a series of gain G values. After acquiring localization frames (e.g. Fig. 9, bottom right), we stacked them and performed MF using u0,0 as a convolution kernel. We computed the Euclidean distance from the highest point of the convolution results to the middle of the frame and legitimized this distance as a point-source localization error. The results are presented in Fig. 11. It can be seen that OPA is able to achieve the precision limit of such a test more than an order of magnitude faster than a CT. Figure 11. View largeDownload slide Comparison of the localization error of a point source using the speckle maxima method. Green dots denote the efficiency of classic optics. For each G, the results for the classic telescope were computed anew. Blue dots denote the quantum telescope/optical parametric amplification efficiency. Figure 11. View largeDownload slide Comparison of the localization error of a point source using the speckle maxima method. Green dots denote the efficiency of classic optics. For each G, the results for the classic telescope were computed anew. Blue dots denote the quantum telescope/optical parametric amplification efficiency. 7 CONCLUSIONS Our semi-classical model, which is more accurate then previously considered models, shows that the intrinsic parametric amplification noise tends to form random clumps when passing the telescope optics. This effect was not included in the previous models of a QT (Kellerer 2014a,b, 2015; Kurek et al. 2016; Kellerer & Ribak 2016) based on the first quantization. However, such clumps, called light speckles, have been shown experimentally to exist (Mosset et al. 2004). The presence of these clumps means that it is difficult to distinguish the signal from the noise and implies that a different signal analysis procedure should be applied. Further modelling and analyses of such noise in greater detail are needed, because an understanding of this issue is essential for the practical realization of a QT. According to our new results, the centroid search approach will not produce any considerable resolution gain. However, the approach based on localizing the maxima of the speckles – the second signal analysis method we tested – offers about a threefold improvement over classical optics. Moreover, both signal analysis methods were shown to be more efficient in the localization of a point source than classical optics, given the same photon count per measurement. Here again the speckle maxima method is superior to the centroid search approach. In principle, it is possible to improve the performance further with the use of noiseless amplification instead of the OPA. However, this approach has never been tested in astronomy (Ralph & Lund 2009; Zavatta et al. 2011; Marek & Filip 2010). Recently, other techniques competitive to OPA have emerged for overcoming the diffraction limit in far-field imaging (Chrostowski et al. 2017). These possibilities are currently under investigation by our group. Acknowledgements We thank our colleague, Dr R. Demkowicz-Dobrzanski, for useful and inspiring discussions, and Dr A. Popowicz for valuable suggestions concerning signal analysis. We also thank the anonymous referee for useful and constructive comments which led to significant improvements in the manuscript. Funding was provided by the Polish National Science Centre grant no. 2016/21/N/ST9/00375, the European Commission FP7 projects SIQS (grant agreement no. 600645) and PhoQuS@UW (grant agreement no. 316244) co-financed by the Polish Ministry of Science and Higher Education (MNSW: 71501E-338/M/2017) and the Foundation for Polish Science TEAM project ‘Quantum Optical Communication Systems’ co-financed by the European Union under the European Regional Development Fund. Footnotes 1 MF is an efficient detector of the known template in a noisy environment and is widely used in radars or sonars, where weak, well-defined reflected signals have to be detected (Helstrom 1968; Woodward 1953). 2 For a review of EMCCD image formation, see section 2 in Tulloch & Dhillon (2011). 3 The structural similarity index is a measure of image similarity. Its value ranges from −1 to 1 (identical images). For details, see Wang et al. (2004). 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Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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