Fig 1 Scheme of the NIQPR.
Published:31 August 2023,
Published Online:11 July 2023,
Received:17 March 2023,
Revised:21 June 2023,
Accepted:23 June 2023
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Quantum entanglement and squeezing have significantly improved phase estimation and imaging in interferometric settings beyond the classical limits. However, for a wide class of non-interferometric phase imaging/retrieval methods vastly used in the classical domain, e.g., ptychography and diffractive imaging, a demonstration of quantum advantage is still missing. Here, we fill this gap by exploiting entanglement to enhance imaging of a pure phase object in a non-interferometric setting, only measuring the phase effect on the free-propagating field. This method, based on the so-called "transport of intensity equation", is quantitative since it provides the absolute value of the phase without prior knowledge of the object and operates in wide-field mode, so it does not need time-consuming raster scanning. Moreover, it does not require spatial and temporal coherence of the incident light. Besides a general improvement of the image quality at a fixed number of photons irradiated through the object, resulting in better discrimination of small details, we demonstrate a clear reduction of the uncertainty in the quantitative phase estimation. Although we provide an experimental demonstration of a specific scheme in the visible spectrum, this research also paves the way for applications at different wavelengths, e.g., X-ray imaging, where reducing the photon dose is of utmost importance.
Quantum imaging
In particular, given the importance of optical phase measurement, appearing in all the science fields, a considerable effort has been made to exploit quantum entanglement or squeezing for this task. Quantum phase estimation through first-order interference involving the mixing of two optical modes in a linear
Thus, in general, apart from some remarkable exceptions
We mention that a fully classical interferometric scheme, exploiting homodyne detection have been recently used to retrieve wide-field phase information with low illumination
Other phase-imaging methods born in the quantum domain exploit second-order intensity correlation measurement (or two-photon coincidence) among signal and idler beams of SPDC to retrieve the phase information. In contrast, the first-order intensity measurement of either the signal or the idler arm does not show interference
Here, we present a quantitative non-interferometric quantum-enhanced phase-imaging (NIQPI) scheme exploiting quantum correlations that do not belong to any of the techniques mentioned above since it does not involve neither interference nor measurements of second-order momenta of the joint photon number distribution. In fact, only first-order momenta (intensities) in both branches are measured, so the full-field phase retrieval is obtained in real-time by quasi-single-shot measurement, as described in the following. We will demonstrate, theoretically and experimentally that, thanks to the quantum correlations, the method can provide a clear advantage compared to the corresponding classical direct imaging at a fixed number of photons interacting with the sample.
The NIQPI protocol exploits the scheme depicted in
1
Fig 1 Scheme of the NIQPR.
Two correlated beams labeled signal (s) and idler (i) are generated by the spontaneous parametric down conversion (SPDC) pumped by a CW laser @405 nm and propagate through an imaging system composed of two lenses (L1 is the far field lens with focal length F = 1 cm and L2 is the imaging lens with focal length of 3 cm) and a test object. An interference filter (IF) is used to select a bandwidth of 40 nm around the degenerate wavelength (@810 nm) and to block the pump. L2 images the far field plane on the camera chip with a magnification factor of about 8. The object is placed near to the far field of the source, and only the probe beam interacts with it. Phase information can be retrieved
where the derivative is approximated by the finite difference of two measurements out of focus,
TIE is experimentally easy and computationally efficient as compared to the conventional phase retrieval techniques and, under suitable hypotheses described in the M&M, the method leads to a unique and quantitative wide-field image of the phase profile
NIQPI can work with partially coherent light and has some advantages compared to interferometric schemes: it can be directly applied to wide-field transmission microscopy settings and it is intrinsically more stable than an interferometric setup
In our experiment, the number of photons per pixel per frame is about n ≈ 103, so that for the purpose of this work we can substitute the continuous quantity I(x) appearing in Eq. (1) with the number of photons detected by the pixel at the coordinate x. Actually, before the TIE algorithm, we apply an averaging filter of size d = 4 to the intensity image, that consists in replacing the count in each pixel by the average count of a neighborhood of size 4 × 4 pix2 around it, so that the final image conserves the same number of pixels. However, the averaging filter does not have any influence on the classical reconstructions, neither positive nor negative, while it improves the quantum reconstruction (see discussion in M&M and related
Fig 2 Sample.
Pure phase objects used in the experiment are sketched
Fig 3 Experimental reconstruction of the "π" sample as a function of the defocusing distance.
First row presents the phase reconstruction when 100 intensity patterns are used. Second and third rows show the single-frame reconstructions for the classical and the quantum case, respectively. The classical reference is realized by the measurement performed using only the signal beam of the SPDC source. The size of each image is 80 × 80 pix2
Fig 4 Pearson correlation between reconstructed and reference images.
The light-blue and yellow curves are the result of a Fourier optics based
Fig 5 Phase estimation.
a The estimated value of the phase step (average of the rectangular selected region) is plotted at different defocusing distances. Experimental points for the classical (yellow dot) and the quantum (purple dot) phase retrieval are compared with the simulations. Simulation line-width and experimental uncertainty bar correspond to one standard deviation after an average over 100 reconstructions. For comparison, we also report the nominal value, estimated by the profilometer in reflection, of the phase step difference between the etched/non-etched areas. b The uncertainty in the phase estimation for quantum and classical cases demonstrate the quantum advantage
Fig 6 Single-frame reconstruction of the squares pattern.
a Examples of classical and quantum reconstructions of the sample with squares in
Fig 7 Intensity images at different defocus.
At dz = 0 the intensity-induced changes are negligible, while structures become visible for displacement of dz = ±0.1 with respect to the conjugate plane. The color-bar refers to the number of photon per pixel
Fig 8
Noise reduction factor (NRF) and Fano factor as a function of the averaging filter size
Fig 9 Pearson correlation as a function of averaging filter size.
The purple (yellow) dots represent the values corresponding to the quantum (classical) experimental reconstructions. The quantum (classical) confidence bands at one standard deviation are also shown in turquoise (yellow)
It is essential to point out that the SPDC source operates in the regime of very low photon number per spatio-temporal mode. In this limit, the photon statistics follows a Poisson distribution (see M&M Sec. for details). So, aside from the negligible contribution of electronic readout noise, the measurement on the single beam is shot-noise limited.
We image pure phase objects reported in
In order to take advantage of the quantum correlations, here we propose to replace into the Eq. (1) the single beam intensity with the following one
2
where
3
where 0 ≤ η ≤ 1 is the heralding efficiency, namely the probability of detecting an idler photon in the pixel in xi conditioned to the detection of the correlated signal photon in the pixel in xs (see M&M section). The parameter α is the average fraction of photons that deviate from the original path due to the phase object and depends on the average phase gradient. It can be experimentally evaluated as the spatial average of the quantity
The third row of
A quantitative analysis of the quality of the reconstructions and of the quantum advantage can be performed by evaluating the Pearson correlation coefficient between the reference phase image and the reconstructed one. The Pearson coefficient is defined as,
4
where
Besides the correct reconstruction of the complex phase profile assessed by the correlation coefficient, in many cases, it is of utmost importance to achieve a quantitative estimation of the phase.
We have also tested a different object, the pattern of regular squares represented in
In summary, these results demonstrate, for the first time, a significant advantage of quantum phase imaging, that can be further extended in the future with various potentially significant applications.
Here, we have demonstrated a genuine quantum enhancement in non-interferometric quantitative phase imaging, showing that the spatially multimode quantum correlations can be used to reduce the detrimental effect of quantum noise in phase reconstruction. The present NIQPI scheme exports the classical methods known as the transport of intensity equation to the quantum regime, which provides real-time wide-field phase imaging and the quantitative local estimation of the phase. The last aspect is fundamental for many applications, providing reliable information on the object's internal parameters related to the phase.
We point out that, compared to the imaging of an amplitude object
A non-interferometric method
Using energy conservation considerations, this equation has been proven valid even with partially coherent sources
In our experiment the validity conditions for the TIE mentioned above, i.e., the paraxial approximation and the partial coherence, are satisfied.
Following the analysis in
5
The noise is assumed independent in the two planes +δz and −δz, so it has been combined in quadrature. The Eq. (5) can be solved by taking the Fourier transform on both sides, leading to
6
where the tilde indicate the Fourier transform and q is the spatial frequency. The damping factor |q|2 of the higher frequencies at the denominator of Eq. (6) and the fact that the quantum noise (shot noise) has a flat white spectrum
Source
In the experiment, we use SPDC in the low gain regime in which a photon of the pump beam (p) (CW laser @405 nm), thanks to the interaction with a bulk beta-barium borate non-linear crystal as long as 15 mm, have a small probability of converting in a couple of photons, usually called signal (s) and idler (i), subject to conservation of energy,
For a Gaussian distributed pump with angular full-width-half-maximum (FWHM) of Δq the spatial cross-correlation is also Gaussian with FWHM of
Test sample
The structures are etched on to a fused Silica glass window (WG41010-A, Thorlabs) with an anti-reflection coating on one side. The window is coated with positive PMMA resist and the design is exposed using electron beams. The exposed structures are developed using a MIBK-IPA solution. After development, the window is submerged in a buffered oxide etch for 30 seconds to etch the structures into the window. The etch depth is determined by the submergence time. The unexposed resist is then removed using acetone solution.
The sample is fabricated to have negligible absorption difference between signal and idler beams. In fact, the two beams pass through the same glass and the only difference between the two beams path is the 66 nm extra glass that introduces a negligible transmission difference (of the order of 10−8(ref.
Detection
We measure the SPDC emission and the effect of the phase object by imaging the far field of the source at the sensor of a CCD camera (Princeton Pixis 1024 Excelon BR) operated in the conventional linear mode. Each pixel delivers a count proportional to the number of incident photons. The proportionality coefficient (electronic gain g) has been carefully estimated (g = 0.97) and the quantum efficiency of the camera is nominally above 95% @810 nm. The electronic readout noise is 4e−/(pix·frame). The number of photons detected per pixel per frame is 103, where the integration time of the camera is set to 100 ms. Thus, the photon flux per pixel is 104 photons/(pix·s).
Because of the finite cross-correlation area defined in the previous section of the M&M, in order to collect most of the correlated photons, two symmetrically placed detectors (or pixels) must have areas larger than the cross-coherence area. Pixel size is 13 μm and a binning of 3 × 3 is performed to set the resolution to 5 μm at the object plane, considering the magnification of about 8x, which matches the measured cross-coherence area. Actually, the heralding efficiency η, i.e., the probability of detecting an idler photon conditioned to the prior detection of the twin signal photon, depends on the pixel size L and possible misalignment Δ of the two pixels compared to the optimal positions, according to this expression:
7
where, η0 is the single photon detection efficiency. As the the pixel size L increases with respect to the coherence area Δx, we have that
As a consequence of that, in Eq. (3), the noise reduction depends on the pixel size used for the measurement. This trade-off between the quantum advantage and the spatial resolution of the intensity measurement has been reported and analyzed in the context of sub-shot-noise imaging of amplitude objects
In the experiment, in order to increase the heralding efficiency, and thus the quantum enhancement, we apply an averaging filter to the intensity image that substitutes the count in each pixel by the average of a square of size d × d pix2 in which the original pixel is on the left-up corner. The quantum correlations are then enhanced because the effective integration area is larger, while the number of pixels in the final image is unvaried. The photon number correlation between the signal and idler beams can be evaluated by measuring the noise reduction factor (NRF) defined as
We underline that in the present imaging of pure phase objects, a certain resolution loss in the intensity pattern is well tolerated. In fact, as it is described in the first section of this M&M, the solution of the TIE tends by itself to suppress the higher frequency component of the intensity perturbation. Thus, to some extent, a reduction of resolution in the intensity measurement does not affect the phase reconstruction of the classical scheme. In
According to the scheme in
8
where
In order to reduce spatial intensity fluctuation, we replace in the TIE the quantity in Eq. (8) with the one in Eq. (2) involving the idler measurement.
The optimal factor kopt appearing there is chosen to minimize the residual fluctuation, by imposing
9
According to the Poisson distribution of the detected photon, we can replace the variance of the intensities appearing in Eq. (9) with the respective quantum mean values. In particular, by performing the spatial averaging, one gets
10
11
The last equality is justified again using the Poisson hypothesis, and introducing the heralding efficiency η that spoils the otherwise perfect signal-idler correlation. By using Eq. (11), and the Poisson hypothesis above, we can rewrite Eq. (9) as,
This work has received funding by the project 20FUN02 POLight of the EMPIR program co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation program, from the European Union's Horizon 2020 Research and Innovation Action under Grant Agreement Qu-Test (HORIZON-CL4-2022-QUANTUM-05-SGA) and from the Project Trapezio QuaFuPhy of San Paolo bank foundation. S.S. would like to acknowledge Dr. Iman E. Zadeh for his supervision for the sample fabrication.
G.O. and I.R.B. devised the scheme of NIQPI. P.B. participated in the realization of the setup and preliminary measurement with G.O., while A.P. and C.N. performed the final experimental acquisitions, which were realized in the laboratories of the research sector directed by M.G. The samples have been prepared and characterized by S.S. and S.P. Simulations and data analysis have been carried out by G.O. and A.P. with the help of I.R.B. I.R.B. and M.G. supervised the whole project. I.R.B. wrote the paper with the contribution of all authors.
All data needed to evaluate the conclusions are reported in the paper. Further data, for reproducibility of the results, are open access available at https://doi.org/10.5281/zenodo.8039275.
The authors declare no competing interests.
Ruo-Berchera, I.&Degiovanni, I. P. Quantum imaging with sub-poissonian light: challenges and perspectives in optical metrology.Metrologia56, 024001 (2019).. [Baidu Scholar]
Moreau, P. -A., Toninelli, E., Gregory, T.&Padgett, M. J. Imaging with quantum states of light.Nat. Rev. Phys.1, 367 (2019).. [Baidu Scholar]
Genovese, M. Real applications of quantum imaging.J. Opt.18, 073002 (2016).. [Baidu Scholar]
Degen, C. L., Reinhard, F.&Cappellaro, P. Quantum sensing.Rev. Mod. Phys.89, 035002 (2017).. [Baidu Scholar]
Pirandola, S., Bardhan, B. R., Gehring, T., Weedbrook, C.&Lloyd, S. Advances in photonic quantum sensing.Nat. Photon.12, 724 (2018).. [Baidu Scholar]
Petrini, G. et al. Is a quantum biosensing revolution approaching? perspectives in nv-assisted current and thermal biosensing in living cells.Adv. Quant. Technol.3, 2000066 (2020).. [Baidu Scholar]
Aasi, J. et al. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light.Nat. Photon.7, 613 (2013).. [Baidu Scholar]
Pradyumna, S. T. et al. Twin beam quantum-enhanced correlated interferometry for testing fundamental physics.Commun. Phys.3, 104 (2020).. [Baidu Scholar]
Taylor, M. A.&Bowen, W. P. Quantum metrology and its application in biology.Phys. Rep.615, 1 (2016).. [Baidu Scholar]
Casacio, C., Madsen, L.&Terrasson, A. Quantum-enhanced nonlinear microscopy.Nature594, 201–206 (2021).. [Baidu Scholar]
Petrini, G. et al. Nanodiamond–quantum sensors reveal temperature variation associated to hippocampal neurons firing.Adv. Sci.9, 2202014 (2022).. [Baidu Scholar]
Schwartz, O.&Oron, D. Improved resolution in fluorescence microscopy using quantum correlations.Phys. Rev. A85, 033812 (2012).. [Baidu Scholar]
Gatto Monticone, D. et al. Beating the abbe diffraction limit in confocal microscopy via non-classical photon statistics.Phys. Rev. Lett.113, 143602 (2014).. [Baidu Scholar]
Samantaray, N., Ruo-Berchera, I., Meda, A.&Genovese, M. Realization of the first sub-shot-noise wide field microscope.Light Sci. Appl.6, e17005(2017).. [Baidu Scholar]
Tenne, R. et al. Super-resolution enhancement by quantum image scanning microscopy.Nat. Photon.13, 116 (2019).. [Baidu Scholar]
Lawrie, B. J., Lett, P. D., Marino, A. M.&Pooser, R. C. Quantum sensing with squeezed light.ACS Photon.6, 1307 (2019).. [Baidu Scholar]
Lee, C. et al. Quantum plasmonic sensors.Chem. Rev.121, 4743 (2021).. [Baidu Scholar]
Polino, E., Valeri, M., Spagnolo, N.&Sciarrino, F. Photonic quantum metrology.AVS Quant. Sci.2, 024703 (2020).. [Baidu Scholar]
Demkowicz-Dobrzanski, R., Jarzyna, M.&Kolodynski, J. Quantum limits in optical interferometry.Prog. Opt.60, 345–435 (2015).. [Baidu Scholar]
Chekhova, M. V.&Ou, Z. Y. Nonlinear interferometers 698 in quantum optics.Adv. Opt. Photon.8, 104 (2016).. [Baidu Scholar]
Hudelist, F. et al. Quantum metrology with parametric amplifier-based photon correlation interferometers.Nat. Commun.5, 3049 (2014).. [Baidu Scholar]
Ono, T., Okamoto, R.&Takeuchi, S. Quantum entanglement-enhanced microscope.Nat. Commun.4, 2426 (2013).. [Baidu Scholar]
Israel, Y., Rosen, S.&Silberberg, Y. Supersensitive polarization microscopy using noon states of light.Phys. Rev. Lett.112, 103604 (2014).. [Baidu Scholar]
Caves, C. M. Quantum-mechanical noise in an interferometer.Phys. Rev. D.23, 1693 (1981).. [Baidu Scholar]
Xiao, M., Wu, L. -A.&Kimble, H. J. Precision measurement beyond the shot-noise limit.Phys. Rev. Lett.59, 278 (1987).. [Baidu Scholar]
Schnabel, R. Squeezed states of light and their applications in laser interferometers.Phys. Rep.684, 1 (2017).. [Baidu Scholar]
Gatto, D., Facchi, P.&Tamma, V. Heisenberg-limited estimation robust to photon losses in a mach-zehnder network with squeezed light.Phys. Rev. A105, 012607 (2022).. [Baidu Scholar]
Kalashnikov, D. A., Paterova, A. V., Kulik, S. P.&Krivitsky, L. A. Infrared spectroscopy with visible light.Nat. Photon.10, 98 (2016).. [Baidu Scholar]
Manceau, M., Leuchs, G., Khalili, F.&Chekhova, M. Detection loss tolerant supersensitive phase measurement with an SU(1, 1) interferometer.Phys. Rev. Lett.119, 223604 (2017).. [Baidu Scholar]
Frascella, G. et al. Wide-field SU(1, 1) interferometer.Optica6, 1233 (2019).. [Baidu Scholar]
Lemos, G. B. et al. Quantum imaging with undetected photons.Nature512, 409 (2014).. [Baidu Scholar]
Töpfer, S. et al. Quantum holography with undetected light.Sci. Adv.8, eabl4301 https://www.science.org/doi/pdf/10.1126/sciadv.abl4301 (2022).. [Baidu Scholar]
Camphausen, R. et al. A quantum-enhanced wide-field phase imager.Sci. Adv.7, eabj2155 https://www.science.org/doi/pdf/10.1126/sciadv.abj2155 (2021).. [Baidu Scholar]
Wolley, O. et al. Near single-photon imaging in the shortwave infrared using homodyne detection.Proc. Natl Acad. Sci. USA120, e2216678120 (2023).. [Baidu Scholar]
Gatti, A., Brambilla, E., Bache, M.&Lugiato, L. A. Correlated imaging, quantum and classical.Phys. Rev. A70, 013802 (2004).. [Baidu Scholar]
Strekalov, D. V., Sergienko, A. V., Klyshko, D. N.&Shih, Y. H. Observation of two-photon "ghost" interference and diffraction.Phys. Rev. Lett.74, 3600 (1995).. [Baidu Scholar]
Valencia, A., Scarcelli, G., D'Angelo, M.&Shih, Y. Two-photon imaging with thermal light.Phys. Rev. Lett.94, 063601 (2005).. [Baidu Scholar]
Zhang, D., Zhai, Y. -H., Wu, L. -A.&Chen, X. -H. Correlated two-photon imaging with true thermal light.Opt. Lett.30, 2354 (2005).. [Baidu Scholar]
Shapiro, J. H.&Boyd, R. W. The physics of ghost imaging.Quant. Inf. Process.11, 949 (2012).. [Baidu Scholar]
Meda, A. et al. Magneto-optical imaging technique for hostile environments: The ghost imaging approach.Appl. Phys. Lett.106, 262405, https://doi.org/10.1063/1.4923336 (2015).. [Baidu Scholar]
Vinu, R. V., Chen, Z., Singh, R. K.&Pu, J. Ghost diffraction holographic microscopy.Optica7, 1697 (2020).. [Baidu Scholar]
Devaux, F., Mosset, A., Bassignot, F.&Lantz, E. Quantum holography with biphotons of high Schmidt number.Phys. Rev. A99, 033854 (2019).. [Baidu Scholar]
Defienne, H., Ndagano, B., Lyons, A.&Faccio, D. Polarization entanglement-enabled quantum holography.Nat. Phys.17, 591 (2021).. [Baidu Scholar]
Aidukas, T., Konda, P. C., Harvey, A. R., Padgett, M. J.&Moreau, P. -A. Phase and amplitude imaging with quantum correlations through fourier ptychography.Sci. Rep.9, 10445 (2019).. [Baidu Scholar]
Hodgson, H., Zhang, Y., England, D.&Sussman, B. Reconfigurable phase contrast microscopy with correlated photon pairs.Appl. Phys. Lett.122, 034001, https://doi.org/10.1063/5.0133980 (2023).. [Baidu Scholar]
Erkmen, B. I.&Shapiro, J. H. Signal-to-noise ratio of gaussian-state ghost imaging.Phys. Rev. A79, 023833 (2009).. [Baidu Scholar]
Brida, G. et al. Systematic analysis of signal-to-noise ratio in bipartite ghost imaging with classical and quantum light.Phys. Rev. A83, 063807 (2011).. [Baidu Scholar]
Morris, P. A., Aspden, R. S., Bell, J. E. C., Boyd, R. W.&Padgett, M. J. Imaging with a small number of photons.Nat. Commun.6, 5913 (2015).. [Baidu Scholar]
Dixon, P. B. et al. Quantum ghost imaging through turbulence.Phys. Rev. A83, 051803 (2011).. [Baidu Scholar]
Bina, M. et al. Backscattering differential ghost imaging in turbid media.Phys. Rev. Lett.110, 083901 (2013).. [Baidu Scholar]
Teague, M. R. Deterministic phase retrieval: a Green's function solution.J. Opt. Soc. Am.73, 1434 (1983).. [Baidu Scholar]
Paganin, D.&Nugent, K. A. Noninterferometric phase imaging with partially coherent light.Phys. Rev. Lett.80, 2586 (1998).. [Baidu Scholar]
Zuo, C. et al. Transport of intensity equation: a tutorial.Opt. Lasers Eng.135, 106187 (2020).. [Baidu Scholar]
Brida, G., Genovese, M.&Ruo Berchera, I. Experimental realization of sub-shot-noise quantum imaging.Nat. Photon.4, 227 (2010).. [Baidu Scholar]
Ruo-Berchera, I. et al. Improving resolution-sensitivity trade off in sub-shot noise quantum imaging.Appl. Phys. Lett.116, 214001, https://doi.org/10.1063/5.0009538 (2020).. [Baidu Scholar]
Ortolano, G., Ruo-Berchera, I.&Predazzi, E. Quantum enhanced imaging of nonuniform refractive profiles.Int. J. Quantum Inf.17, 1941010, https://doi.org/10.1142/S0219749919410107 (2019).. [Baidu Scholar]
Lu, C. -H., Reichert, M., Sun, X.&Fleischer, J. W. Quantum phase imaging using spatial entanglement.https://arxiv.org/abs/1509.01227(2015). [Baidu Scholar]
Borodin, D., Schori, A., Zontone, F.&Shwartz, S. X-ray photon pairs with highly suppressed background.Phys. Rev. A94, 013843 (2016).. [Baidu Scholar]
Sofer, S., Strizhevsky, E., Schori, A., Tamasaku, K.&Shwartz, S. Quantum enhanced x-ray detection.Phys. Rev. X9, 031033 (2019).. [Baidu Scholar]
Moreau, P. -A. et al. Demonstrating an absolute quantum advantage in direct absorption measurement.Sci. Rep.7, 6256 (2017).. [Baidu Scholar]
Losero, E., Ruo-Berchera, I., Meda, A., Avella, A.&Genovese, M. Unbiased estimation of an optical loss at the ultimate quantum limit with twin-beams.Sci. Rep.8, 7431 (2018).. [Baidu Scholar]
Avella, A., Ruo-Berchera, I., Degiovanni, I. P., Brida, G.&Genovese, M. Absolute calibration of an emccd camera by quantum correlation, linking photon counting to the analog regime.Opt. Lett.41, 1841 (2016).. [Baidu Scholar]
Paganin, D., Barty, A., McMahon, P. J.&Nugent, K. A. Quantitative phase-amplitude microscopy. ⅲ. The effects of noise.J. Microsc.214, 51 (2004). https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.0022-2720.2004.01295.x.. [Baidu Scholar]
Meda, A. et al. Photon-number correlation for quantum enhanced imaging and sensing.J. Opt.19, 09400 (2017).. [Baidu Scholar]
Khashan, M.&Nassif, A. Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2–3 µm.Opt. Commun.188, 129 (2001).. [Baidu Scholar]
Gunjala, G.&Waller, L.Open Source PhaseGUI(UC Berkeley, 2014). [Baidu Scholar]
Voelz, D. G.Computational Fourier Optics: a Matlab Tutorial(SPIE, 2011).https://doi.org/10.1117/3.858456.. [Baidu Scholar]
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